We construct a classical oracle proving that, in a relativized setting, the set of languages decidable by an efficient quantum verifier with a quantum witness (QMA) is strictly bigger than those decidable with access only to a classical witness (QCMA). The separating classical oracle we construct is for a decision problem we coin spectral Forrelation -- the oracle describes two subsets of the boolean hypercube, and the computational task is to decide if there exists a quantum state whose standard basis measurement distribution is well supported on one subset while its Fourier basis measurement distribution is well supported on the other subset. This is equivalent to estimating the spectral norm of a "Forrelation" matrix between two sets that are accessible through membership queries. Our lower bound derives from a simple observation that a query algorithm with a classical witness can be run multiple times to generate many samples from a distribution, while a quantum witness is a "use once" object. This observation allows us to reduce proving a QCMA lower bound to proving a sampling hardness result which does not simultaneously prove a QMA lower bound. To prove said sampling hardness result for QCMA, we observe that quantum access to the oracle can be compressed by expressing the problem in terms of bosons -- a novel "second quantization" perspective on compressed oracle techniques, which may be of independent interest. Using this compressed perspective on the sampling problem, we prove the sampling hardness result, completing the proof.

翻译：我们构造了一个经典谕示，证明在相对化设定下，由高效量子验证者借助量子证据（QMA）可判定的语言集合严格大于仅能访问经典证据（QCMA）可判定的语言集合。我们构造的用于分离的经典谕示针对一个我们称为谱傅里叶关联的判定问题——该谕示描述了布尔超立方体的两个子集，其计算任务是判断是否存在一个量子态，使其在标准基下的测量分布能很好地支撑于一个子集，同时在其傅里叶基下的测量分布能很好地支撑于另一个子集。这等价于估计两个可通过成员查询访问的集合之间的“傅里叶关联”矩阵的谱范数。我们的下界源于一个简单的观察：使用经典证据的查询算法可被多次运行以从某个分布中生成多个样本，而量子证据是一种“一次性使用”的对象。这一观察使得我们可以将证明QCMA下界的问题简化为证明一个采样困难性结果，该结果不会同时证明QMA下界。为了证明QCMA的该采样困难性结果，我们观察到通过将问题用玻色子表示——一种新颖的压缩谕示技术的“二次量子化”视角，可能具有独立研究价值——可以压缩对谕示的量子访问。利用这一对采样问题的压缩视角，我们证明了采样困难性结果，从而完成了证明。