Collapses in quantum-classical probabilistically checkable proofs and the quantum polynomial hierarchy

TL;DR

This paper explores the structure of quantum proof systems, demonstrating collapse results and the robustness of uniqueness constraints, while establishing quantum analogues of classical complexity theorems and analyzing entanglement's role.

Contribution

It extends classical theorems to quantum settings, proving collapse results for quantum hierarchies and analyzing the impact of entanglement and uniqueness on quantum proof complexity.

Findings

UniqueQCPCP equals QCPCP under certain reductions

Quantum Karp-Lipton theorem analogue established

Collapse of bounded-entanglement quantum hierarchy above level four

Abstract

We investigate the structure of quantum proof systems by establishing collapse results that reveal simplifications in their complexity landscape. By extending classical theorems such as the Karp-Lipton theorem to quantum settings and analyzing uniqueness in quantum-classical PCPs, we clarify how various constraints influence computational power. Our main contributions are: (1) We show that restricting quantum-classical PCPs to unique proofs does not reduce their power: $\mathsf{UniqueQCPCP} = \mathsf{QCPCP}$ under $\mathsf{BQ}$-operator and randomized reductions. This parallels the known $\mathsf{UniqueQCMA} = \mathsf{QCMA}$ result, indicating robustness of uniqueness even in quantum PCP-type systems. (2) We prove a non-uniform quantum analogue of the Karp-Lipton theorem: if $\mathsf{QMA} \subseteq \mathsf{BQP}/\mathsf{qpoly}$, then $\mathsf{QPH} \subseteq \mathsf{Q\Sigma}_2/\mathsf{qpoly}$. This conditional collapse suggests limits on quantum advice for $\mathsf{QMA}$-complete problems. (3) We define a bounded-entanglement version of the quantum polynomial hierarchy, $\mathsf{BEQPH}$, and prove that it collapses above the fourth level. We also introduce the separable hierarchy $\mathsf{SepQPH}$ (zero entanglement), for which the same collapse result holds. These collapses stem not from entanglement, as in prior work, but from the convex structure of the protocols, which renders higher levels tractable. Collectively, these results offer new insights into the structure of quantum proof systems and the role of entanglement, uniqueness, and advice in defining their complexity.