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: UniqueQCPCP = QCPCP under BQ-operator and randomized reductions. This parallels the known UniqueQCMA = 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 QMA BQP/qpoly, then QPH QΣ₂/qpoly. This conditional collapse suggests limits on quantum advice for QMA-complete problems. (3) We define a bounded-entanglement version of the quantum polynomial hierarchy, BEQPH, and prove that it collapses above the fourth level. We also introduce the separable hierarchy 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.
Anand et al. (Tue,) studied this question.