We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the $q$-local Hamiltonian problem with constant promise gap can be solved in $\mathsf{QCMA}$, then $\mathsf{QPCP}[q] \subseteq \mathsf{QCMA}$ for any $q \in O(1)$. (iii) If $\mathsf{QMA}(k)$ has a quantum PCP for any $k \leq \text{poly}(n)$, then $\mathsf{QMA}(2) = \mathsf{QMA}$, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exists (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

翻译：我们定义了量子PCP的一般表述，该表述捕获了适应性与多个非纠缠证明者，并给出了到具有常数承诺间隙的局部哈密顿量的量子归约的详细构造。该归约被证明是证明量子PCP性质的多功能子程序，使我们能够展示：(i) 当证明查询次数为常数时，非适应性量子PCP可以模拟适应性量子PCP。事实上，即使非适应性量子PCP仅从所有可能索引组合的子集中均匀随机选取证明索引，这一结论仍然成立，从而回答了Aharonov、Arad、Landau和Vazirani（STOC '09）提出的开放问题。(ii) 如果具有常数承诺间隙的$q$-局部哈密顿量问题可在$\mathsf{QCMA}$中求解，则对于任意$q \in O(1)$有$\mathsf{QPCP}[q] \subseteq \mathsf{QCMA}$。(iii) 若对于任意$k \leq \text{poly}(n)$，$\mathsf{QMA}(k)$具有量子PCP，则$\mathsf{QMA}(2) = \mathsf{QMA}$，这一结果连接了量子复杂性理论中两个长期存在的开放问题。此外，我们还证明了存在某些（量子）谕示，使得特定量子PCP断言为假。因此，任何证明量子PCP猜想的尝试——正如经典PCP定理的情况一样——都需要（量子）非相对化技术。