Quantum Coordination Advantages in AI State-Tracking Tasks: Semantic Compilation and Latent Memory

This paper proves inference-time quantum coordination advantages for specified AI state-tracking tasks. A solver compresses semantic history into a future-accessible boundary state and later answers a query. The paper counts communication B, persistent instance-dependent memory M, and local work D; classical recurrence, caches, tools, and recomputation are allowed and charged.

The central result is a boundary-preserving semantic-compilation theorem. It maps a finite one-way, streaming, or adaptive causal task into a semantic AI interface while preserving event order and access to past input. Classical boundary-state lower bounds and quantum-memory upper bounds transfer up to explicit compiler overhead, independently of the finite-precision recurrent architecture.

Two applications have classical semantics. Matched-entity synopsis QA inherits the hidden-matching separation between O(log N) qubits and Ω(√N) classical boundary bits. Continual requirements auditing inherits a Max-kSAT streaming separation: a recurrent solver uses O(log5 n log(1/δ)) qubits and polylogarithmic classical workspace to obtain a 0.7172-approximation, whereas every classical one-pass finite-information solver attaining that ratio requires Ω(√n) coordination width.

As a quantum-native compiler test, a stabilizer latent-state dialogue uses n qubits, while every exact finite-state classical causal online realization satisfies B + M ≥ ½n2 + (3/2 − log23)n + O(1). The source protocols, streaming algorithms, and stabilizer witness are imported; the new result is their architecture-independent semantic transfer. These are memory and coordination separations, not runtime or empirical advantages for present-day language models. The stabilizer result assumes exact simulation and ideal noiseless quantum memory.

The paper defines a boundary-relative coordination cost framework. A computational boundary Σt separates the event that has processed history ht from the event that must respond to condition ct. B is explicit information crossing the boundary, M is internal state retained across the boundary, and D is local processing depth after the boundary. The generative coordination region CostϵA(P; Σ) is the set of (B, M, D) such that some A-generator simulates P within error ϵ. The boundary is part of the model; the prompt or context counts as communication only when it carries information from one modeled computational event to another.

The paper distinguishes general classical causal simulators from restricted chart simulators. A general classical causal simulator may use randomized, context-dependent response kernels and adaptive state updates; its complete future-accessible boundary configuration λ must contain all past-dependent information available after the cut, but the transition and response rules themselves are unrestricted. For a static past–future probability table FΣ(P)u,v = Pr(U = u, V = v), the general separator theorem gives B + M ≥ log2 rank+ FΣ(P). For an online process with Hankel table HP, the corresponding quantity is the causal positive-realization rank. A restricted chart simulator instead requires every counted transcript–memory state to select a context-independent global response chart; its covering-number bound applies only to that cover-admissible class. Neither a chart covering lower bound nor a KWB support-counting lower bound applies to a general causal simulator without an explicit reduction.

A finite classical latent-state generator has hidden state λt ∈ Λ. The past history prepares a distribution over hidden states, wt(λ) = w(λht), and the current query ct is answered by a response kernel R(otct, λt). Thus q(otht, ct) = Σλt wt(λtht)R(otct, λt). After observing ct, ot, the state may update by another stochastic kernel U(λt+1λt, ct, ot). Classical state complexity KclD,ϵ(P; Σ) is the minimum number of classical hidden coordination states needed by a depth-D classical generator to simulate P within error ϵ relative to Σ. Quantum state complexity Kqϵ(P; Σ) is the minimum Hilbert-space dimension needed by a quantum latent-state generator. Proposition 1 gives the basic state-count lower bound: if a simulator with B bits of explicit boundary communication and M bits of retained memory can select among at most 2(B+M) effective classical coordination states, then exact or ϵ-approximate simulation of P implies B + M ≥ log2 KclD,ϵ(P; Σ). Proposition 2 shows that if P has a finite classical sufficient state zt ∈ Z such that P(otht, ct) = P(otzt, ct) and zt+1 is sampled from a kernel depending only on (zt, ct, ot), then KclD,0(P; Σ) ≤ Z and MclD,0(P; Σ) ≤ ⌈log2Z⌉. This elementary upper bound is the reason that transformer failures alone do not prove a quantum advantage; a recurrent classical model may repair the failure by spending memory.

Classical repairs are framed as resource moves: recurrence and state-space models primarily increase M; scratchpads and chain-of-thought traces are B-type resources; external memory and tools mix communication and memory; latent thinking and iterative inference increase D; long context is not automatically a cost—if it is supplied as part of the task input, it is data, but if it is generated or compressed by the model to coordinate future events, it is part of the resource accounting.

A quantum latent-state generator carries a density operator ρt ∈ D(Ht). For each current condition ct, the generator uses a POVM Moct o and outputs p(otht, ct) = Tr(Moctt ρt). The latent state then updates by a quantum instrument, ρt+1 = Ect,ot(ρt). The important distinction is not merely that ρt is continuous or random; it is that different queries ct may correspond to noncommuting measurements. A classical hidden state can answer all queries by carrying a large enough table of counterfactual responses. A latent-state tracking task is noncommuting if there are query contexts c, c′ whose associated measurements cannot be jointly represented as coarse-grainings of one fixed classical response variable without increasing the hidden coordination state space. Proposition 3 gives the contextual coordination lower bound: if P induces a finite query model Eh and every depth-D representation in a classical model class C within error ϵ needs at least K effective states, then any depth-D implementation in C with boundary resources (B, M) must satisfy B + M ≥ log2 K.

Theorem 1 gives the coordination-cost separation criterion. Let Pn be a family of generative processes and Cn a named classical simulator class. Suppose there is a quantum latent-state generator for Pn with Hilbert-space dimension dn, and every depth-Dn generator in Cn simulating Pn within error ϵn needs at least Ln hidden coordination states. Then every such implementation in Cn satisfies B + M ≥ log2 Ln, whereas the quantum implementation uses at most ⌈log2 dn⌉ qubits of latent memory. The separation is linear, polynomial, or exponential according to the growth of log2 Ln − log2 dn.

The paper discusses related work. Prior work on quantum generative models shows that quantum correlations can increase expressive power. Quantum predictive-memory separations show that quantum states can reduce the memory needed to simulate certain stochastic processes. An expressivity advantage asks which distributions can be represented compactly; a coordination advantage asks how many states, messages, or update steps are needed to preserve one latent process. Sequential contextuality was formulated as a classical internal-memory cost by Kleinmann et al., and Fagundes and Kleinmann extended that analysis to the full probabilistic Peres–Mermin correlations. Karanjai, Wallman, and Bartlett obtained growing stabilizer-simulation memory bounds. Prakash converts graph-theoretic contextuality into an exponential quantum-memory advantage for a formal-language promise problem. Gao et al. show that quantum correlations can give compact generative representations outside the reach of selected classical Bayesian-network and neural-network families; their hidden-Markov-model result has the shape classical memory = Ω((log D)2), quantum memory = O(log D). Teo et al. define strong k-contextuality for translation tasks and show that a strongly k-contextual task cannot be represented to finite relative entropy by a classical streaming model with fewer than k latent states. The present paper does not claim the contextuality-to-classical-memory link as new; rather, it recasts such separations as boundary-relative coordination statements and connects them to modern state-tracking failures.

Definition 7 defines a boundary-preserving semantic compiler. Let Πn be a finite task with an ordered sequence of environment events, solver actions, and computational boundaries specified by an access model An. A boundary-preserving semantic compiler maps each source event online to a finite text, symbolic, or multimodal block and maps solver outputs back to the source output alphabet. It must preserve the event order, adaptive choices, and source boundaries; never re-supply a past source event after its boundary unless that record is explicitly charged as persistent state; preserve the source acceptance relation or transcript distribution up to error ηn; and use at most an bits of parser, renderer, and compiler workspace, including every compiler record retained across a source boundary. The resulting semantic task is denoted SemAn(Πn).

Theorem 2 is the semantic coordination transfer. Let SemAn be a boundary-preserving semantic compiler with workspace an, compilation error ηn, and at most Tn event boundaries. Then: (1) every classical solver for SemAn(Πn) with error at most ϵ and peak coordination width WΣ satisfies WΣ ≥ CclA,ϵ+ηn(Πn) − an − O(log Tn); (2) if Πn has a quantum solver with error at most ϵ and resource profile (qn, cn), then the compiled semantic task has a solver with error at most ϵ + ηn using qn qubits and cn + an + O(log Tn) classical bits across the corresponding boundaries. For an exact eventwise compiler that retains no additional instance-dependent state, exact classical causal-state lower bounds and quantum-memory upper bounds are preserved without asymptotic loss. The proof composes a classical semantic solver with the online encoder, parser, and output decoder; because the compiler preserves event order and does not reintroduce expired source records, the composition is a valid An-solver for Πn. At each source boundary its complete state consists of the semantic solver's state, at most an compiler bits, and O(log Tn) event-counter bits. For the quantum direction, run the semantic parser online, apply the source quantum channel or measurement selected by the decoded event, retain its qn-qubit state and cn-bit classical state, and render the source output semantically.

Transformer state-tracking failures provide the motivating classical bottleneck. A feed-forward transformer can often use the context as a workaround, but persistent dynamic state is not free. The coordination-cost picture turns engineering choices into resource moves: recurrence maps to M, scratchpad or tool messages map to B, extra latent computation maps to D. The latent-state-persistence tasks of Huang et al. are useful because they separate local linguistic plausibility from the ability to preserve a hidden state across many queries; in this paper they play the role of a classical stress test, not a quantum benchmark. The topological state-tracking problems emphasized by Mozer, Siddiqui, and Liu provide a second useful shell; the route to a quantum separation is to keep their dialogue structure while replacing the tracked state by a noncommuting latent state.

For relational state tracking, if a history determines N = Θ(n2) independent pairwise facts among n entities and a query asks for any selected fact, then an exact classical tracker needs Θ(n2) retained bits. However, this observation alone does not imply a quantum advantage. An m-qubit latent representation that can answer any one of the N independent facts with success probability p > 1/2 is a quantum random-access code, so Nayak's bound gives m ≥ (1 − H2(p))N = Ω(n2). Thus an arbitrary classical relation table cannot be compressed to O(n) qubits if the benchmark allows reliable random access to all its entries. The plausible quantum target is narrower: the text-induced IO relation should have high classical coordination rank but low quantum, or positive semidefinite, rank. In binary-output form, a family of histories and queries defines a nonnegative matrix Ah,c = P(o = 1h, c). A classical latent-state factorization corresponds to a nonnegative factorization of A, while a quantum latent-state representation has the form Ah,c = Tr(Ecρh), which is a positive-semidefinite factorization. A natural-text or text-wrapped relational benchmark would therefore show a coordination advantage only if its conditional-response matrix has large nonnegative rank but small PSD rank.

Definition 8 defines entity-attribute synopsis QA. A family consists of finite sets XN of passage states, CN of query contexts, and ON of answers, together with a relation RN ⊆ XN × CN × ON. A passage hx is an unambiguous text encoding of an entity-attribute state x ∈ XN. A query c ∈ CN is revealed after the passage has been processed. A valid answer is any o ∈ ON such that (x, c, o) ∈ RN. The benchmark boundary is essential: the passage is read first, then only a boundary state is retained, and the query is revealed later. If the full passage is carried across the boundary, its length is charged to B; if a classical model rereads or rescans the passage after seeing the query, that repair is charged to D.

Corollary 1 is the one-way lift to synopsis QA. Suppose the relation problem RN, under a chosen input distribution, admits a one-way quantum protocol with qN qubits and success probability at least 1 − ϵ, while every one-way randomized classical protocol with the same success probability requires at least cN bits. Then the corresponding entity-attribute synopsis QA family has a quantum state-tracking solver using qN qubits across the passage–query boundary, and every classical solver in the same one-way boundary model satisfies B + M ≥ cN. The proof is the exact one-boundary specialization of Theorem 2, with the passage and query as the two source events and no retained compiler state. The quantum upper bound is obtained by running the quantum one-way encoder after parsing the passage and retaining its qN-qubit message as the boundary state. Conversely, any classical QA solver using m = B + M boundary bits gives a one-way classical protocol for RN: Alice parses x, forms the passage hx, runs the passage processor, and sends the resulting boundary state to Bob; Bob parses c, runs the query responder, and outputs its answer.

The content is not tied to a specific Boolean operation. Direct random access to arbitrary entity attributes is ruled out by the quantum random-access-code obstruction, but any entity-attribute relation family with a one-way quantum/classical separation yields a synopsis-QA separation. Hidden matching is a simple instantiation of this more general lifting principle. In that instantiation, a passage describes N named records, each with a binary attribute such as cohort, stance, access level, or case label. After the passage has been processed, a later query gives a list of disjoint record pairs and asks the model to report any listed pair together with whether the two records have the same or different labels. Under the memory version of this task, the problem is exactly the hidden matching problem: it has an O(log N)-qubit one-way protocol and requires Ω(√N) classical one-way bits at bounded error. This avoids the random-access-code obstruction because the query does not ask for a pre-specified stored bit; it lets the solver choose any edge from a large matching and report the corresponding relation.

The chart interpretation uses the sheaf-theoretic framework for contextuality and its database reading. For a fixed matching M, the response condition is local: output one edge of M and the corresponding parity. A classical boundary state s, however, induces a global response chart gs: M ↦ (i, j, b) over all possible matchings. A solver using B + M classical bits can select at most 2(B+M) such charts after reading the history. The hidden-matching lower bound says that, at bounded error, no small family of classical global charts can cover the required history–query relation. This is not a bare KS contradiction; if the full string x is stored, then the assignment bij = xi ⊕ xj is a perfectly good global parity chart for all pairs. The obstruction is resource-sensitive: classical simulation must spend many bits to select an adequate chart, while the quantum protocol keeps a compact phase state from which a query context extracts one valid local relation by interference.

Definition 9 defines matched-entity consistency QA. Let N be even. A passage hx describes N named entities with binary labels x ∈ 0, 1 N, using a fixed unambiguous grammar. A query cM presents a perfect matching M on the entity set [N]. A valid answer is any triple (i, j, b) such that (i, j) ∈ M and b = xi ⊕ xj. Here b = 0 means that the two selected entities have the same label, and b = 1 means that they have different labels.

Corollary 2 gives the matched-entity QA separation. Consider the one-way state-tracking protocol in which x is drawn uniformly from 0, 1 N, the passage processor sees hx, a boundary state is retained, and only then a uniformly random perfect matching M is revealed to the query responder. The query responder must output a valid triple for (x, M) with probability at least 2/3. There is an exact quantum boundary-state protocol using ⌈log2 N⌉ qubits. Any bounded-error classical protocol in the same one-way boundary model requires B + M = Ω(√N) bits. For the quantum upper bound, after reading the passage prepare ψx⟩ = (1/√N) Σi (−1) xi i⟩, which uses ⌈log2 N⌉ qubits. Given a matching M, measure first in the orthogonal decomposition span i⟩, j⟩, (i, j) ∈ M, which selects an edge (i, j) ∈ M. Conditional on this edge, the state is proportional to (−1) xi i⟩ + (−1) xⱼ j⟩. Now measure in the basis (i⟩ ± j⟩)/√2 inside that two-dimensional subspace. The sign is + iff xi ⊕ xj = 0 and − iff xi ⊕ xj = 1, so the responder outputs (i, j, xi ⊕ xj) with certainty. For the classical lower bound, suppose a classical state-tracking solver uses m = B + M boundary bits and succeeds with probability at least 2/3. This solver gives a one-way randomized communication protocol for hidden matching: Alice, given x, forms the passage hx, runs the passage processor, and sends the resulting m-bit boundary state to Bob; Bob, given M, runs the query responder and outputs its triple. The one-way communication lower bound for hidden matching therefore implies m = Ω(√N).

RNNs and state-space models do not trivialize the question. They do trivialize one weak claim: it is not enough to show that a fixed-depth feed-forward transformer loses track of a latent variable. A recurrent model can store the variable. But recurrence changes the resource point from small M to larger M. The nontrivial question is whether, for some process family Pn, every classical recurrent repair requires Ω(f(n)) bits while a quantum latent state uses O(g(n)) qubits with g(n) ≪ f(n).

Definition 10 defines peak online coordination width. Consider a solver that consumes update blocks x1,..., x T in order. Let Zt contain all stream-dependent information available after xt has been consumed and before x t+1 arrives. This includes retained activations, recurrent states, accessible cache entries, generated scratchpad symbols, and records written to an external tool or store. If Zt has at most 2(w t) operationally distinguishable classical values, define WΣ = max t w t to be the solver's peak online coordination width across the family of cuts Σ = (Σ0,..., Σ T). Under the bit accounting used above, WΣ ≤ max t(Bt + Mt) when every accessible explicit record and internal state is included in Bt + Mt. Conversely, representing Zt by an index costs at most wt bits. A neural state with r real coordinates at p-bit operational precision contributes at most rp bits; allowing an exact real number to encode an unbounded stream would leave the finite-space model and is not a finite-information classical baseline.

Definition 11 defines a semantics-preserving online compiler. Let Πn be a streaming relation problem with update alphabet Un and output relation RΠn ⊆ Un* × Yn. A semantics-preserving online compiler consists of a prefix-decodable encoding Encn(u) of each update as one text or multimodal block and an answer decoder Decn such that (u1:T, y) ∈ RΠn ⇐⇒ (Encn(u1),..., Encn(u T), Decn−1(y)) is accepted. The compiler has overhead an if parsing the current block and rendering the final answer use at most an bits of workspace and retain no additional stream-dependent information between update boundaries. The no-retained-information clause prevents the linguistic wrapper itself from hiding a large database. It does not require constant-length text: entity identifiers may use O(log n) bits, provided only the current record is being parsed.

Corollary 3 is the online semantic lift. Let Scl(n, ϵ) be a lower bound on the space of every randomized one-pass classical streaming algorithm for Πn with error at most ϵ. If an online compiled AI solver has error at most ϵ, compiler overhead an, and peak classical coordination width WΣ, then WΣ ≥ Scl(n, ϵ) − an − O(log T). If Πn has a one-pass quantum streaming algorithm using Sq qubits and Cq classical bits, then the compiled task has a quantum recurrent solver using Sq qubits and Cq + an + O(log T) classical bits. The proof uses Theorem 2 with the one-pass access model and the exact eventwise compiler above.

Remark 4 notes architecture independence: the classical implication uses only the number of distinguishable states carried across update cuts. The update map may be nonlinear, randomized, and computationally unbounded. It therefore applies equally to finite-precision RNNs, nonlinear SSMs, recurrent transformers, KV-cache systems, scratchpads, and tool-using agents, provided all persistent information is counted in WΣ. Extra local depth cannot reconstruct distinctions that were not retained after the stream passed.

Remark 5 discusses fixed parameters versus instance-dependent state. The parameters of a pretrained model are part of the fixed algorithm description and are not charged as online memory. Write a recurrent implementation schematically as z t+1 = Fθ(zt, xt), p(otct, zt) = Gθ(ct, zt). The streaming lower bound already permits Fθ and Gθ to be arbitrarily complicated. Nevertheless, if two realized histories induce the same future-accessible state zt, fixed parameters cannot make their response distributions differ under the same future query. Model weights may store the update rule or a vast read-only lookup table, but the instance-dependent index selecting the realized history must still cross the boundary. Test-time weight updates, adapters, fast weights, or model selection that depend on the stream are therefore part of Zt and are charged to WΣ.

Remark 6 discusses the full-context loophole. If the complete raw transcript remains freely available for random access, the solver is no longer one-pass and Corollary 3 does not apply. One must either charge the stored transcript as external memory and its retrieval as boundary traffic, or analyze a multi-pass model. An L-token context over a vocabulary of size V can itself carry up to L log2 V raw token-index bits, and its accessible KV cache is also stream-dependent state. A sufficiently large context can therefore satisfy the lower bounds in this paper; it is a classical repair with a potentially large WΣ, not a violation of the theorem.

Corollary 4 is the dynamic relation-summary separation. For the terminal approximation ratio 0.4844 and failure probability δ, the dynamic relation-summary task has a quantum recurrent solver using O(log5 n log(1/δ)) qubits of online workspace, plus logarithmic compiler workspace. Every finite-information classical recurrent solver in the same one-pass access regime satisfies WΣ = Ω(√n). Equivalently, its family of effective recurrent coordination states has size Kcl online ≥ 2(Ω(√n)). The proof cites Kallaugher, Parekh, and Voronova's one-pass quantum streaming algorithm with the displayed space bound and approximation ratio. The classical streaming lower bound they invoke states that every ratio strictly larger than 4/9 requires Ω(√n) bits. The fixed relation grammar is prefix-decodable with O(log n) workspace. Corollary 3 transfers both bounds, and 0.4844 > 4/9. This task realizes DeepMind's schematic update st = f(s t−1, xt): each sentence modifies a compact synopsis of a growing relational world. The conclusion is stronger than a failure theorem for a feed-forward transformer; giving the model recurrence repairs the topological depth problem, but every classical repair still needs Ω(√n) peak retained bits at the target approximation ratio.

Definition 12 defines the continual requirements-audit task. Fix k ≥ 2. A task instance contains n named binary decisions and a time-ordered dialogue r1,..., r T. Each requirement utterance rt has a certified semantic parse as a clause Ct containing at most k literals. Once rt has been processed, it is unavailable except through the solver's retained state. On the terminal query, the solver outputs a number Z estimating OPT(C1:T) = max a∈ 0,1 n t: Ct(a) = 1, or equivalently the normalized compliance score OPT(C1:T)/T. The theorem-certified version uses a controlled natural-language grammar, so each requirement can be parsed independently with O(log n) workspace. A benchmark may additionally contain ordinary paraphrases, domain vocabulary, and coreference, but then semantic-parser error is a separate empirical layer. The memory theorem already applies to the exactly parseable subset; a quantum upper bound for the richer surface form additionally assumes a shared online semantic front end.

Corollary 5 is the continual requirements-audit separation. For every fixed k ≥ 2, the controlled-language continual requirements-audit task admits a one-pass quantum recurrent solver which, with probability at least 1 − δ, outputs Z satisfying OPT(C1:T) ≥ Z ≥ 0.7172 OPT(C1:T) using O(log5 n log(1/δ)) qubits of online workspace. It also uses polylogarithmic classical working bits, including O(log n) exact counters. Every finite-information classical recurrent solver attaining that ratio in the same one-pass regime has WΣ = Ω(√n). The proof cites Wang and Yang's one-pass quantum streaming algorithm for Max-kSAT; its quantum sketches also use polylogarithmic classical control and working bits, and its preprocessing retains logarithmic exact counters. The classical streaming lower bound rules out every ratio strictly larger than √2/2 ≈ 0.7071 in o(√n) space. The controlled requirement grammar is a semantics-preserving online compiler with logarithmic workspace, so Corollary 3 preserves both bounds. This is a natural AI state-tracking problem in the operational sense used by Mozer, Siddiqui, and Liu: the accumulated requirement set is an evolving world state, and its task-sufficient synopsis must be updated as st = f(s t−1, rt). The assistant is not asked to recall arbitrary past sentences or output a quantum object; it produces one classical planning diagnostic. The result is stronger than the observation that a transformer may lose track of a satisfying assignment: the lower bound ranges over all bounded-space classical update rules, including recurrent repairs. The task also has a clear limitation: it estimates the optimum compliance value; it does not output the optimizing plan. The cited quantum streaming algorithm does not establish a compact quantum advantage for plan construction, and the present paper does not claim one.

The paper clarifies what is imported and what is new. The Max-DiCut and Max-kSAT quantum algorithms, approximation constants, and classical streaming lower bounds are imported results. Rewording their records as sentences does not create a new quantum algorithm. The new claim developed here is the general transfer principle and the associated AI task model: after fixing an online semantic boundary, a streaming lower bound becomes a lower bound on the peak coordination width of every finite-information recurrent AI implementation, including the standard architectural repairs to transformer state tracking. Continual requirements auditing supplies a practical planning semantics for that theorem. A purported small classical solver must be using uncharged transcript access, unbounded numerical precision, a weaker output guarantee, or a different access model.

The paper also addresses finite-size interpretation. These theorems establish asymptotic coordination separations, not a practical memory saving at ordinary LLM scales. The Ω(√n) bounds hide constants: at n = 106, the scaling term √n is only 103, whereas a 128k-token context over a 105-word vocabulary can carry about 2.1 × 106 raw token-index bits before counting the physical KV cache. Likewise, the explicit stabilizer expression below is about 5.0 × 103 bits at n = 100 qubits. Moreover, an O(log5 n) quantum upper bound need not beat √n at moderate n, especially after constants, fault-tolerance, and interface costs are included. No finite-size crossover or practical quantum-memory advantage is claimed here.

Definition 13 defines the contextual LSP benchmark. For each size parameter n, a contextual latent-state-persistence benchmark consists of: Hn allowed histories, Cn allowed query contexts, ρh: h ∈ Hn latent states prepared by histories, Moc: o ∈ Oc c∈Cn query-dependent output measurements, and Ec,o c,o state-update instruments. At test time the benchmark presents a history ht and a query context ct. The target conditional distribution is Pn(otht, ct) = Tr(Moctt ρht), and after observing ot the latent state updates as ρh t+1 = Ec t,o t(ρht). A model is evaluated by the average total-variation distance, log loss, or success probability of its conditional predictions over an adaptive sequence of histories and queries. The intended boundary is the time cut after ht has been processed but before ct is revealed. If the complete history is re-supplied together with the query, then a classical model may recompute the latent state from the raw transcript; in the present accounting that repair is charged to local depth D, not treated as free state tracking. This definition contains ordinary LSP as the jointly classical special case. If all states and measurements are jointly diagonal in a common basis, then there is a classical sufficient variable zt and Proposition 2 applies. The task becomes contextual in the operational sense used here only when the same history can later be queried in contexts that do not admit a small common response chart.

The operational modification from LSP to contextual LSP is small. In an ordinary hidden-state benchmark, the history prepares a latent variable and later questions ask for facts about that variable. In a contextual benchmark, the history prepares a latent object and later questions choose one of several incompatible tests of that object. A language wrapper could describe the history as a lab notebook, simulation trace, symbolic circuit, or world-state update; the mathematical core is that ct is not just a request for a stored fact, but a measurement context. The resulting classical repair options are still allowed. A classical model may store a chart in recurrent memory, write intermediate chart data into a scratchpad, or recompute a chart after seeing the query. The point is that these repairs now have visible costs: stored chart data maps to M, written chart data maps to B, reconstructed chart data maps to D.

Definition 14 defines stabilizer contextual LSP. The stabilizer contextual-LSP family Pn stab, the benchmark version of the quantum-memory seed imported from Ref. [3], is obtained by taking Hn to be histories of Clifford gates and previous Pauli measurement outcomes on n qubits. Each history prepares an n-qubit stabilizer state ρh. A query context c ∈ Cn is a commuting family of Pauli observables, the output o is the corresponding string of measurement outcomes, and the update map is the usual stabilizer measurement update.

Definition 15 defines a semantic stabilizer dialogue. A semantic stabilizer dialogue is a natural-language or symbolic-language presentation of Pn stab. The transcript describes Clifford updates and previous Pauli measurement outcomes using an unambiguous finite grammar; the next prompt describes a commuting Pauli context; and the required answer is the corresponding outcome distribution or a sample from it. The semantic target process is still Pn stab; the text wrapper only supplies a state-tracking interface of the kind used in transformer state-tracking benchmarks.

Definition 16 defines adaptive-complete recurrent simulation. A classical recurrent implementation of a contextual-LSP family is adaptive-complete if, for every finite adaptive policy that chooses the next query context as a function of the previous history and outcomes, the implementation reproduces the joint distribution of the full transcript. The boundary state at time t consists of the retained recurrent state together with any explicit transcript crossing the chosen boundary Σt. Thus an implementation using resources (B, M) has at most 2(B+M) effective boundary states at each cut.

The paper clarifies the relation to the genuine-global construction. Reference [3] contains two logically distinct steps. Its finite-causal-witness lemma is already a single-system statement about the n-qubit stabilizer seed: it constructs the finite adaptive interface Wn and proves the causal-state lower bound used below. A subsequent, optional flag lift embeds that seed into a genuinely global multipartite model and transfers the same cost by conditioning on the flag. The present LSP benchmark imports only the first step. It therefore needs no k = 1 or single-party reduction from the genuinely global theorem, and its quantum upper bound remains the n-qubit seed realization. Applying the separate flag lift would instead produce a genuinely global restriction with a small additional flag-memory overhead, but that extra physical structure is not used in the AI state-tracking claim.

Lemma 1 is the imported finite causal stabilizer witness. For every n ≥ 2, there is a finite adaptive interface Wn ⊂ Pn stab such that every exact finite-state classical causal online realization of Wn has at least Kn ≥ 2n Πⱼ=1n (2j + 1) / (5 · 3n−2) boundary states. Consequently, every exact adaptive-complete classical recurrent implementation of Pn stab satisfies B + M ≥ log2 Kn. The proof cites Karanjai–Wallman–Bartlett's result that every set of more than mn = 5 · 3n−2 pure n-qubit stabilizer states admits a stabilizer partitioning measurement. For every subset of mn + 1 preparations, include one such measurement and, after each relevant outcome, one allowed single-shot test distinguishing the resulting orthogonal pair. The stabilizer preparation and measurement sets are finite for fixed n, so their union defines a finite interface Wn. If one causal boundary state occurred with positive probability after all preparations in one of these subsets, its common response kernel would assign positive probability to some partitioning outcome and successor state. Two preparations would then reach orthogonal postmeasurement records through that same successor state, while the subsequent distinguishing test requires different certain outcomes, a contradiction. Thus one causal state can occur in the support of at most mn pure preparations. There are 2n Πⱼ=1n (2j + 1) pure stabilizer states, giving the stated count. This finite-witness upgrade from the KWB overlap theorem to arbitrary finite-state causal realizations is exactly the single-system finite causal witness from the stabilizer overlap bound lemma of Ref. [3], restated here to make the reduction self-contained. It precedes, and does not rely on, that reference's genuinely global flag-lift theorem. An exact adaptive-complete implementation of the full process remains exact when restricted to Wn. Its complete future-accessible transcript and retained state have at most 2(B+M) values, so 2(B+M) ≥ Kn.

Corollary 6 is the imported stabilizer latent-state separation. The family Pn stab is exactly generated by a quantum recurrent model using n qubits of latent memory. Any exact adaptive-complete finite-state classical causal recurrent implementation, with all future-accessible boundary information counted, satisfies B + M = Ω(n2). More explicitly, for n ≥ 2, B + M ≥ log2(2n Πⱼ=1n (2j + 1) / (5 · 3n−2)) = ½n2 + (3/2 − log23)n + O(1) = Ω(n2). The proof states that the quantum implementation stores the physical n-qubit stabilizer state and applies the requested Clifford or Pauli-measurement update. Restricting an exact adaptive-complete classical implementation to the finite interface Wn preserves its boundary-state set. Lemma 1 therefore gives the displayed ratio and its logarithm. Expanding log2(2j + 1) = j + log2(1 + 2−ʲ) gives the quadratic asymptotic. The factor in this count is 2j + 1, exponential in j. As a low-dimensional check, the formula gives 23(3)(5)(9) = 1080 pure stabilizer states at n = 3; this factor is what produces the quadratic logarithmic growth.

Corollary 7 is the contextual quantum-AI state-tracking separation. Relative to the boundary after the transcript has been semantically processed and before the next query context is revealed, the text-wrapped stabilizer dialogue is generated by a quantum recurrent model with n qubits of latent memory. Any exact adaptive-complete finite-state classical causal recurrent solver, with all future-accessible semantic boundary information counted, satisfies B + M = Ω(n2). The proof states that the controlled grammar is an exact boundary-preserving semantic compiler: removing the linguistic surface leaves the stabilizer contextual-LSP process Pn stab, each current block is parsed with O(log n) workspace, and no parser record survives the semantic boundary. A quantum solver applies each decoded update to the physical n-qubit latent state. Theorem 2, applied to the imported finite-state lower bound in Lemma 1, gives the stated Ω(n2) classical bound.

Remark 7 notes that this corollary is the sense in which an NLP-style task can carry a provable quantum coordination advantage. The advantage belongs to the noncommuting state-tracking problem preserved by the text, not to ordinary natural language understanding by itself. If the full raw transcript is re-supplied and a classical model recomputes a stabilizer tableau after each prompt, the stored or retransmitted transcript is charged to B + M and its processing to D; neither resource is free.

Remark 8 notes that this benchmark is intentionally quantum-native. Rephrasing it as natural language does not make the advantage a property of ordinary language modeling; it only gives a user-facing wrapper around a noncommuting state-tracking task. Its value is that it isolates the exact technical target: prove the reduction from an adaptive state-tracking interface to causal boundary-state complexity. The finite witness above achieves that reduction exactly; robust approximate and natural-task versions remain open.

The scope of the transferred bound is that the displayed Ω(n2) separation applies to every exact finite-state classical causal online realization of the adaptive-complete interface; it is not restricted to a preselected chart or neural architecture. The finite witness is what upgrades the KWB overlap count to this general causal class. The theorem still does not cover constant-error approximation, a batch algorithm with free random access to the full transcript, or uncounted infinite-precision state. Those are different approximation or access models.

Adaptive completeness matters because a one-step benchmark distribution is weaker than a process simulator. A model might predict the next answer well on a fixed distribution of histories without carrying enough state to answer all future compatible queries. The stabilizer lower-bound route therefore needs an adaptive benchmark: after any history that the model itself has helped generate, an evaluator may choose a new commuting Pauli context and continue. This is the operational content of state tracking, and it is what turns a conditional prediction benchmark into a candidate memory lower-bound problem.

The evaluation protocol is that an evaluator can implement adaptive completeness without inspecting the model's internal state. At the start of each trial it resets the model, supplies an allowed preparation-and-update history, and then selects each next commuting Pauli context as a function of the full interaction transcript. It records the model's response, applies the corresponding target-process update, and continues for a prescribed horizon. Repeated trials estimate the joint distribution of complete adaptive trajectories, which is compared with the target process in total variation or log loss. The policy family must include context choices that distinguish histories merged by a candidate simulator; a fixed i.i.d. test set does not provide this guarantee. Constructing an efficient worst-case policy, or a finite certificate that is complete for a given model class, remains an open algorithmic problem.

The paper includes a classical-baseline audit table summarizing the logical strength of each result. The state-count and separation criteria are conditional on the explicitly named class C. The semantic coordination transfer is general within the named source access model, with all compiler state charged. The synopsis and matched-entity QA results apply to arbitrary randomized one-way protocols. The online semantic lift and its two tasks apply to arbitrary randomized one-pass finite-information update rules. Strong k-contextuality of Teo et al. is interface-limited to finite-state HMM or finite-precision autoregressive realization. The imported stabilizer witness applies to arbitrary exact finite-state causal online realization. The compiled contextual AI dialogue is a general transfer of that result.

The discussion section identifies three different roles for the task families. Classical-looking calibrations (matched-entity QA and continual requirements auditing) instantiate established one-way or streaming separations inside ordinary database and planning semantics; they validate the boundary accounting and the semantic lifts, but their underlying quantum algorithms and lower bounds are imported. In particular, the requirements-audit advantage comes from streaming Max-kSAT and is not claimed to arise from contextuality. The quantum-native compiler test (stabilizer measurement tracking) imports the clean finite witness of Ref. [3]; the new role it plays here is to test whether a semantic state-tracking interface preserves an adaptive causal-state separation. Its limitations are explicit: adaptive completeness is required, the source theorem is exact, and the semantics remain quantum-native unless a certified wrapper is used. Extensions toward natural processes include quantum predictive-memory models providing classical stochastic processes with compact quantum representations; contextual interactive dialogues, scientific time series, and embodied-agent histories are further targets, but difficult state tracking alone does not imply noncommutativity. A convincing extension must identify a concrete query family with a large classical causal-state lower bound and a compact quantum realization.

The open problems are: (1) foundational robustness—identify equivalence classes of computational boundaries under which the B, M, D region is stable, and separate representation or simulation cost from the cost of learning a generator; (2) robust and efficient witnesses—extend the exact stabilizer theorem to ϵ-approximate causal simulation and replace the potentially large finite witness by efficient adaptive evaluator policies and statistical certificates, with semantic wrappers preserving the same complete causal interface including gates, measurements, randomness, and updates; (3) finite and physical advantage—determine constants, crossover scales, matching classical upper bounds, and actual context or KV-cache costs, with a physical comparison also including coherence time, error correction, refresh, and classical–quantum input/output overhead over the full stream; (4) natural tasks and outputs—find noncommuting latent-state processes closer to practical generation while retaining provable causal-state lower bounds, including classical-looking interactive benchmarks, natural-language variants with controlled parser error, and strengthening the requirements-audit result from value estimation to construction of an approximately optimal plan.

Based on this paper, I can improve AI systems in the following specific ways:

1. Implement a Resource-Aware State-Tracking Module

• The improved AI system can explicitly track and budget its own memory (M), communication (B), and computation depth (D) across computational boundaries.

• It can detect when it is about to exceed classical memory limits on state-tracking tasks and flag this to the user or switch to a quantum-assisted mode if available.

2. Add a Classical Coordination-Width Estimator

• The system can estimate its peak online coordination width (WΣ) for any given task by counting distinguishable retained states, cache entries, and scratchpad symbols.

• It can then predict whether it will fail on tasks with known lower bounds (e.g., matched-entity QA with Ω(√N) classical bits) and proactively warn the user or allocate more resources.

3. Build a Boundary-Aware Compiler for Semantic Tasks

• The system can automatically compile streaming tasks into semantic interfaces while preserving event order and access to past input, using the boundary-preserving semantic compiler framework.

• It can ensure that no past source event is re-supplied after its boundary unless explicitly charged as persistent state, preventing hidden full-context loopholes.

4. Enable Quantum-Classical Hybrid Execution

• For tasks with proven quantum advantages (e.g., stabilizer dialogues, matched-entity QA, continual requirements auditing), the system can automatically switch to a quantum latent-state generator when a quantum processor is available.

• It can use O(log N) qubits for hidden-matching-based tasks and O(log5 n log(1/δ)) qubits for Max-kSAT streaming tasks, while maintaining classical output.

5. Implement a Noncommutativity Detector

• The system can analyze a given task's conditional-response matrix to determine if it has high nonnegative rank but low PSD rank, indicating a potential quantum coordination advantage.

• It can then recommend quantum-native implementations for such tasks and avoid wasting classical memory on them.

6. Add Adaptive-Completeness Verification

• The system can run adaptive evaluation protocols that choose query contexts based on previous responses, rather than fixed i.i.d. test sets.

• This allows it to verify whether a model truly tracks latent state or merely predicts well on a fixed distribution, catching cheating models that lack sufficient internal state.

7. Create a Classical-Repair Cost Calculator

• The system can automatically compute the cost of classical repairs (recurrence, scratchpads, external memory, recomputation) for any state-tracking failure.

• It can then choose the cheapest repair strategy or determine that no classical repair is feasible within resource limits, triggering a quantum fallback.

8. Implement a Contextual LSP Benchmark Generator

• The system can generate contextual latent-state-persistence benchmarks where histories prepare latent objects and queries choose incompatible tests (e.g., noncommuting Pauli measurements).

• This allows systematic evaluation of any AI system's ability to handle noncommuting state-tracking tasks, with clear resource accounting.

9. Add a Finite-Size Crossover Predictor

• The system can compute the exact crossover scale where quantum memory advantages become practically relevant, accounting for constants, fault-tolerance overhead, and interface costs.

• It can then advise users whether to use quantum or classical approaches for their specific problem size (e.g., n = 106 vs. n = 1012).

10. Build a Semantic Stabilizer Dialogue Interface

• The system can parse natural-language descriptions of Clifford gates and Pauli measurements into a symbolic stabilizer state-tracking task.

• It can then execute the quantum updates on n qubits and generate natural-language responses about measurement outcomes, providing a user-friendly interface to a provably quantum-advantaged task.