QMA vs. QCMA and Pseudorandomness

TL;DR

This paper explores the potential separation between quantum and classical proof systems using oracles, linking it to quantum pseudorandomness conjectures and their implications for quantum advantage.

Contribution

It establishes that a classical oracle separating QMA from QCMA exists if a certain quantum pseudorandomness conjecture holds, connecting complexity theory with quantum pseudorandomness.

Findings

Existence of a classical oracle separating QMA from QCMA under a pseudorandomness conjecture

Equivalence between oracle separation and quantum advantage in distribution distinguishing

Insight into the power of quantum proofs over classical proofs

Abstract

We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating $\mathsf{QMA}$ from $\mathsf{QCMA}$. Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called "dense" distributions. Our result can be viewed as establishing a "win-win" scenario: either there is a classical oracle separation of $\mathsf{QMA}$ from $\mathsf{QCMA}$, or there is quantum advantage in distinguishing pseudorandom distributions on permutations.