QMA vs. QCMA and Pseudorandomness

TL;DR

The paper tackles whether a classical oracle can separate the quantum proof class $\\mathsf{QMA}$ from the classical-guess verifiable class $\\mathsf{QCMA}$, proposing a conditional result: assuming a quantum pseudorandomness conjecture for dense permutation distributions, there exists a classical oracle $F$ with $\\mathsf{QMA}^F \neq \\mathsf{QCMA}^F$. The approach centers on a graph-oracle model that encodes either a disconnected component structure or a highly expanding graph, called the Components problem, and analyzes how quantum vs classical witnesses fare under polylogarithmic query budgets. A key innovation is connecting the QCMA upper bounds/ lower bounds to a decomposition of high-entropy permutation sources into dense components, then invoking the conjecture to argue that fixed-coordinate information cannot rescue a QCMA verifier; in addition, an unconditional separation is established via an interactive game framework, strengthening the overall narrative. The work thus ties fundamental questions about quantum vs classical proofs to pseudorandomness and cryptographic-type hardness, with potential implications for post-quantum security and the power of quantum proofs in structured oracle settings.

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.

Figures (2)

• Figure 1: The algorithm $M$ in the proof of \ref{['lem:bqpred']}.
• Figure 2: This figure illustrates the iterative $\mathrm{Reconnect}_{\rho}$ procedure. The first layer represents all Yes instances conditioned on $A \in S$ and $B \in \bar{S}$, and each circle represents a graph in the corresponding set. There is an edge from a graph $G$ in layer $i$ to a graph $G'$ in layer $i+1$ if $\varphi(G)=G'$ . Each graph in every layer has the same number of incoming edges from the previous layer. Thus the figure is illustrating that the iterative $\mathrm{Reconnect}_{\rho}$ procedure starts with a uniform distribution over $\mathcal{L}_{\mathrm{yes}}(A,B)$, and transforms it into uniform distributions over smaller and smaller sets, with the last set being $\mathcal{L}_{\mathrm{yes}}^{\rho}.$

Theorems & Definitions (66)

• Theorem 1.1: Main theorem, informal
• Conjecture 1.2: Simplified quantum pseudorandomness conjecture
• Conjecture 1.3: Aaronson-Ambainis conjecture aaronson2009need
• Lemma 1.3: Weaker version of \ref{['conj:simplified']}
• Conjecture 1.3: Quantum pseudorandomness conjecture for random permutations
• Proposition 1
• Lemma 2.1
• proof
• Corollary 2.2
• proof