Non-signaling proofs, motivated by quantum computation, have found applications in cryptography and hardness of approximation. An important open problem is characterizing the power of no-signaling proofs. It is known that 2-prover no-signaling proofs are characterized by PSPACE, and that no-signaling proofs with poly(n)-provers are characterized by EXP. However, the power of k-prover no-signaling proofs, for 2kpoly(n) remained an open problem. We show that k-prover no-signaling proofs (with negligible soundness) for k=O(√logn) are contained in PSPACE. We prove this via two different routes that are of independent interest. In both routes we consider a relaxation of no-signaling called sub-no-signaling. Our main technical contribution (which is used in both our proofs) is a reduction showing how to convert any sub-no-signaling strategy with value at least 1−2−Ω(k2) into a no-signaling one with value at least 2−O(k2). In the first route, we show that the classical prover reduction method for converting k-prover games into 2-prover games carries over to the no-signaling setting with the following loss in soundness: if a k-player game has value less than 2−ck2 (for some constant~c>0), then the corresponding 2-prover game has value at most 1−2dk2 (for some constant~d>0). In the second route we show that the value of a sub-no-signaling game can be approximated in space that is polynomial in the communication complexity and exponential in the number of provers.
Non-signaling proofs, motivated by quantum computation, have found applications in cryptography and hardness of approximation. An important open problem is characterizing the power of no-signaling proofs. It is known that 2-prover no-signaling proofs are characterized by PSPACE, and that no-signaling proofs with poly(n)-provers are characterized by EXP. However, the power of k-prover no-signaling proofs, for 2kpoly(n) remained an open problem. We show that k-prover no-signaling proofs (with negligible soundness) for k=O(√logn) are contained in PSPACE. We prove this via two different routes that are of independent interest. In both routes we consider a relaxation of no-signaling called sub-no-signaling. Our main technical contribution (which is used in both our proofs) is a reduction showing how to convert any sub-no-signaling strategy with value at least 1−2−Ω(k2) into a no-signaling one with value at least 2−O(k2). In the first route, we show that the classical prover reduction method for converting k-prover games into 2-prover games carries over to the no-signaling setting with the following loss in soundness: if a k-player game has value less than 2−ck2 (for some constantc>0), then the corresponding 2-prover game has value at most 1−2dk2 (for some constantd>0). In the second route we show that the value of a sub-no-signaling game can be approximated in space that is polynomial in the communication complexity and exponential in the number of provers.
Non-Signaling Proofs with O(logn) Provers are in PSPACE
Non-signaling proofs, motivated by quantum computation, have found applications in cryptography and hardness of approximation. An important open problem is characterizing the power of no-signaling proofs. It is known that 2-prover no-signaling proofs are characterized by PSPACE, and that no-signaling proofs with poly(n)-provers are characterized by EXP. However, the power of k-prover no-signaling proofs, for 20), then the corresponding 2-prover game has value at most 1−2dk2 (for some constant~d>0). In the second route we show that the value of a sub-no-signaling game can be approximated in space that is polynomial in the communication complexity and exponential in the number of provers.
We construct a 1-round delegation scheme (i.e., argument-system) for every language computable in time t = t ( n ), where the running time of the prover is poly ( t ) and the running time of the verifier is n · polylog ( t ). In particular, for every language in P we obtain a delegation scheme with almost linear time verification. Our construction relies on the existence of a computational sub-exponentially secure private information retrieval ( PIR ) scheme. The proof exploits a curious connection between the problem of computation delegation and the model of multi-prover interactive proofs that are sound against no-signaling (cheating) strategies , a model that was studied in the context of multi-prover interactive proofs with provers that share quantum entanglement, and is motivated by the physical principle that information cannot travel faster than light. For any language computable in time t = t ( n ), we construct a multi-prover interactive proof ( MIP ), that is, sound against no-signaling strategies, where the running time of the provers is poly ( t ), the number of provers is polylog ( t ), and the running time of the verifier is n · polylog ( t ). In particular, this shows that the class of languages that have polynomial-time MIP s that are sound against no-signaling strategies, is exactly EXP . Previously, this class was only known to contain PSPACE . To convert our MIP into a 1-round delegation scheme, we use the method suggested by Aiello et al. (ICALP, 2000), which makes use of a PIR scheme. This method lacked a proof of security. We prove that this method is secure assuming the underlying MIP is secure against no-signaling provers.
We describe two generic ways to make multi-prover classical games resistant against entangled provers. The first uses quantum communication and a quantum verifier, the second adds an additional prover. This leads to several new results on the power of proof systems with entangled provers. We show that NEXP ⊆ QMIP1,s(2, 1) and NEXP ⊆ MIP1,s(3, 1) with soundness s = 1− 2− poly(n) and PSPACE ⊆ MIP1,s(2, 1) with soundness s = 1 − 1/ poly(n), providing the first non-trivial bounds in this setting. Moreover, our results imply that, unless P = NP, the value of entangled prover games cannot be computed by semi-definite programs that are polynomial in the size of the verifier’s system, a method that has been successful for more restricted quantum games. ∗also at CNRS & LRI, Univerite de Paris-Sud, Orsay, France †Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848 and by an Alon Fellowship of the Israeli Higher Council of Academic Research. ‡Part of this work was completed at Caltech. Supported by the National Science Foundation under Grants PHY-0456720 and CCF-0524828, by EU project QAP, and by NWO VICI project 639-023-302. Part of this research has been funded by the Dutch BSIK/BRICKS project. §Work partly done while at LRI, Univ. de Paris-Sud, Orsay.
Interactive proof systems form an important complexity model that has been central to many prominent results in computational complexity theory, such as those on probabilistically checkable proofs, hardness of approximation, and fundamental cryptographic primitives. In this thesis, we study the quantum-enhanced version of interactive proof systems, in which each party has access to quantum computing resources. We focus on its power and limitations, which lead us to the following results. (*) We provide an alternative and conceptually simpler proof of Jain et al.’s recent breakthrough result, which demonstrates that the expressive power of quantum interactive proof systems coincides with that of its classical counterpart, and therefore PSPACE. (*) We determine the complexity of several variants of quantum competing-prover interactive proof systems, which introduce zero-sum games into interactive proof systems. In particular, we prove that the complexity class of two-turn quantum refereed games coincides with PSPACE, answering an important open problem of Jain et al.’s. Our result suggests that a more general model called double quantum interactive proof systems coincides with PSPACE, which subsumes and unifies all previous results on short refereed games. (*) We contribute to the study of two-prover quantum Merlin-Arthur games (QMA(2)), which exhibit an intriguing tradeoff between an intrinsic quantum ingredient, entanglement, and computational power. We prove a PSPACE upper bound for a variant of QMA(2) that is to date the most general one known in PSPACE. We also provide a quasi-polynomial time algorithm for optimizing linear functions over separable states, giving an alternative to the one proposed by Brandao et al. Our main technical contribution behind the above results is the equilibrium value method for obtaining space-efficient algorithms for a class of optimization problems that arise naturally in quantum computation. We also contribute to QMA-complete problems, where we demonstrate that the “Non-Identity Check” problem remains QMA-complete on circuits of poly-logarithmic depths, improving upon polynomial depths from previous results.
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1- epsilon, promised that one of the two is the case. Furthermore, as long as epsilon is small enough, this result does not depend on the gap epsilon. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap epsilon decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIP^{co}(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1-s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIP^{co}_{delta}(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1-s gets arbitrarily small. While we do not know any computable time upper bound on the class MIP^{co}, a result of the first author and Vidick shows that for s = 1-1/poly(f(n)) and delta = 1/poly(f(n)), the class MIP^{co}_delta(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.