Interactive Proofs

Abstract: We investigate two resources whose effects on quantum interactive proofs remain poorly understood: the promise of unentanglement, and the verifier’s ability to condition on an intermediate measurement, which we call post-measurement branching. We first show that unentanglement can dramatically increase computational power: three-round unentangled quantum interactive proofs equal NEXP, even if only the first message is quantum. By contrast, we prove that if the verifier uses no post-measurement branching, then the same type of unentangled proof system has at most the power of QAM. Finally, we investigate post-measurement branching in two-round quantum-classical proof systems. Unlike the equivalence between public-coin and private-coin classical interactive proofs, we give evidence of a separation in the quantum setting that arises from post-measurement branching.

Abstract: While the role of entanglement in quantum proof systems has been extensively studied, the computational power of unentanglement remains poorly understood. Since entanglement admits many inequivalent multipartite structures, it is natural to ask how more fine-grained structural promises affect computational power. In this work we investigate a mild promise: each proof is internally separable, meaning that after tracing out one register, a designated constant-size subsystem is separable from the rest—even though the overall proof may still be entangled across every bipartition. We prove a qualitative jump from one proof to two: with one internally separable proof, the resulting class is contained in $\EXP$ (even allowing inverse-exponential completeness–soundness gap), whereas with two unentangled internally separable proofs, the class equals $\NEXP$ at constant gap. Notably, in the $\NEXP$ construction, the second proof is used solely to implement a SWAP-based purity test.