Quantum entanglement is a fundamental property of quantum mechanics and plays a crucial role in quantum computation and information. We study entanglement via the lens of computational complexity by considering quantum generalizations of the class NP with multiple unentangled quantum proofs, the so-called QMA(2) and its variants. The complexity of QMA(2) is a longstanding open problem, and only the trivial bounds QMA $\subseteq$ QMA(2) $\subseteq$ NEXP are known. In this work, we study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote $\text{QMA}^+(2)$. In this setting, we are able to design proof verification protocols for problems both using logarithmic size quantum proofs and having a constant probability gap in distinguishing yes from no instances. In particular, we design global protocols for small set expansion, unique games, and PCP verification. As a consequence, we obtain NP $\subseteq \text{QMA}^+_{\log}(2)$ with a constant gap. By virtue of the new constant gap, we are able to ``scale up'' this result to $\text{QMA}^+(2)$, obtaining the full characterization $\text{QMA}^+(2)$=NEXP by establishing stronger explicitness properties of the PCP for NEXP. One key novelty of these protocols is the manipulation of quantum proofs in a global and coherent way yielding constant gaps. Previous protocols (only available for general amplitudes) are either local having vanishingly small gaps or treat the quantum proofs as classical probability distributions requiring polynomially many proofs thereby not implying non-trivial bounds on QMA(2). Finally, we show that QMA(2) is equal to $\text{QMA}^+(2)$ provided the gap of the latter is a sufficiently large constant. In particular, if $\text{QMA}^+(2)$ admits gap amplification, then QMA(2)=NEXP.

翻译：量子纠缠是量子力学的基本特性，并在量子计算与量子信息中扮演关键角色。我们通过计算复杂性的视角研究纠缠，考虑NP类的量子推广——即具有多个无纠缠量子证明的QMA(2)及其变体。QMA(2)的复杂性是一个长期未解难题，目前仅知其平凡边界QMA ⊆ QMA(2) ⊆ NEXP。本文研究非负振幅无纠缠量子证明的威力，将其记为类QMA^+(2)。在此设定下，我们能够设计针对问题的验证协议，这些协议既使用对数规模量子证明，又能在区分是与非实例时保持恒定概率间隙。具体而言，我们为小集扩张、唯一游戏及PCP验证设计了全局协议。由此，我们得到具有恒定间隙的NP ⊆ QMA^+_log(2)。借助新的恒定间隙，我们通过为NEXP的PCP建立更强的显式性质，将这一结果“放大”至QMA^+(2)，从而获得完整刻画QMA^+(2)=NEXP。这些协议的一个关键创新在于以全局相干方式操控量子证明，从而实现恒定间隙。先前的协议（仅适用于一般振幅）要么是局部且间隙可忽略地小，要么将量子证明视为经典概率分布而需要多项式数量证明，从而无法推出QMA(2)的非平凡边界。最后，我们证明若QMA^+(2)的间隙为充分大的常数，则QMA(2)=QMA^+(2)。特别地，若QMA^+(2)允许间隙放大，则QMA(2)=NEXP。