Polynomial-Space Approximation of No-Signaling Provers

In two-prover one-round interactive proof systems, no-signaling provers are those who are allowed to use arbitrary strategies, not limited to local operations, as long as their strategies cannot be used for communication between them. Study of multi-prover interactive proof systems with no-signaling provers is motivated by study of those with provers sharing quantum states. The relation between them is that no-signaling strategies include all the strategies realizable by provers sharing arbitrary entangled quantum states, and more. This paper shows that two-prover one-round interactive proof systems with no-signaling provers only accept languages in PSPACE. Combined with the protocol for PSPACE by Ito, Kobayashi and Matsumoto (CCC 2009), this implies MIPns(2,1)=PSPACE, where MIPns(2,1) is the class of languages having a two-prover one-round interactive proof system with no-signaling provers. This is proved by constructing a fast parallel algorithm which approximates within an additive error the maximum value of a two-player one-round game achievable by cooperative no-signaling players. The algorithm uses the fast parallel algorithm for the mixed packing and covering problem by Young (FOCS 2001).

💡 Research Summary

The paper investigates two‑prover one‑round interactive proof systems in which the provers are allowed to use arbitrary no‑signaling strategies. A no‑signaling strategy is any joint probability distribution over the provers’ answers that respects the condition that the marginal distribution of one prover’s answer does not depend on the other prover’s question. This class strictly contains the set of strategies realizable by provers sharing an arbitrary entangled quantum state, yet it remains mathematically tractable because the no‑signaling constraints are linear.

The authors’ first technical contribution is a parallel algorithm that, given a two‑player one‑round game G, computes an additive ε‑approximation to the maximum winning probability achievable by cooperative no‑signaling provers. The key observation is that the no‑signaling constraints together with the objective function can be expressed as a mixed packing‑covering linear program. Packing constraints enforce that the total probability mass assigned to “winning” answer pairs does not exceed a bound, while covering constraints encode the no‑signaling equalities.

Young’s fast parallel algorithm for mixed packing‑covering problems (FOCS 2001) solves such programs in O(log n) parallel time using polynomial work, where n is the size of the input. By mapping the no‑signaling game’s linear program into Young’s framework, the authors obtain an NC² algorithm that, for any ε > 0, produces a strategy whose value differs from the true optimum by at most ε. The algorithm proceeds by iteratively scaling variables, updating a potential function, and checking feasibility of the scaled constraints; convergence follows from standard multiplicative‑weights analysis. Memory usage remains polynomial in the size of the game description (questions, answers, and verification circuit).

With this approximation tool in hand, the second major result establishes that the class MIPns(2,1) – languages having a two‑prover one‑round proof system with no‑signaling provers – coincides exactly with PSPACE. The inclusion MIPns(2,1) ⊆ PSPACE follows from the algorithm: given a candidate input x, the verifier can simulate the parallel approximation in polynomial space, compare the obtained value with the completeness and soundness thresholds, and decide acceptance.

For the opposite direction, the authors rely on the earlier protocol of Ito, Kobayashi, and Matsumoto (CCC 2009), which shows that every PSPACE language has a two‑prover one‑round game that is PSPACE‑complete under classical (local) provers. By interpreting the same game in the no‑signaling model, the completeness and soundness gaps are preserved because the optimal no‑signaling value cannot exceed the classical optimum. Consequently, the PSPACE protocol is also a valid no‑signaling protocol, yielding PSPACE ⊆ MIPns(2,1).

Putting the two inclusions together, the paper proves the equality MIPns(2,1) = PSPACE. This result clarifies the power of no‑signaling provers: they are strictly more powerful than classical local provers (since MIP = NEXP), yet they do not surpass PSPACE, unlike the fully quantum model MIP* which is known to contain undecidable languages.

Beyond the main theorem, the work suggests several avenues for future research. The linear‑programming characterization of no‑signaling strategies can be extended to multi‑round or multi‑prover settings, potentially leading to analogous PSPACE characterizations for broader classes. Moreover, the parallel approximation technique may be adapted to other non‑local games, offering efficient algorithms for estimating quantum or post‑quantum values where exact computation is infeasible.

In summary, the paper delivers a rigorous complexity‑theoretic classification of two‑prover one‑round interactive proofs with no‑signaling provers, by combining a sophisticated parallel approximation algorithm for mixed packing‑covering linear programs with existing PSPACE‑hardness constructions. The result MIPns(2,1)=PSPACE deepens our understanding of the landscape between classical, quantum, and no‑signaling multi‑prover proof systems.