Succinct arguments for QMA from standard assumptions via compiled nonlocal games
Tony Metger, Anand Natarajan, Tina Zhang
TL;DR
The paper tackles the challenge of giving a succinct, classical-verifier argument system for $\mathsf{QMA}$ under standard cryptographic assumptions. It combines a KLVY-style compiler (to convert two-prover nonlocal games into a single-prover cryptographic protocol) with cryptographic tools like collapsing hash functions and a light form of quantum homomorphic encryption to achieve succinctness without relying on indistinguishability obfuscation. The authors develop nonuniform Gowers-Hatami and small-bias techniques to robustly self-test Pauli operators and to manage mixed-vs-pure basis measurements within compiled games, including a subsampling strategy for Hamiltonians to maintain a constant-gap $\mathsf{QMA}$ problem. Their framework yields constant-round, polylogarithmic-communication classical verification for $\mathsf{QMA}$ with a quantum prover, and they further promote a fully succinct version via LMS extraction and CMSZ extraction. Overall, the work broadens the baseline for succinct quantum verification by showing that standard assumptions suffice and highlighting a unifying path between nonlocal-game techniques and cryptographic delegation for quantum verification.
Abstract
We construct a succinct classical argument system for QMA, the quantum analogue of NP, from generic and standard cryptographic assumptions. Previously, building on the prior work of Mahadev (FOCS '18), Bartusek et al. (CRYPTO '22) also constructed a succinct classical argument system for QMA. However, their construction relied on post-quantumly secure indistinguishability obfuscation, a very strong primitive which is not known from standard cryptographic assumptions. In contrast, the primitives we use (namely, collapsing hash functions and a mild version of quantum homomorphic encryption) are much weaker and are implied by standard assumptions such as LWE. Our protocol is constructed using a general transformation which was designed by Kalai et al. (STOC '23) as a candidate method to compile any quantum nonlocal game into an argument system. Our main technical contribution is to analyze the soundness of this transformation when it is applied to a succinct self-test for Pauli measurements on maximally entangled states, the latter of which is a key component in the proof of MIP*=RE in quantum complexity.