A distribution testing oracle separation between QMA and QCMA

It is a long-standing open question in quantum complexity theory whether the definition of $\textit{non-deterministic}$ quantum computation requires quantum witnesses $(\textsf{QMA})$ or if classical witnesses suffice $(\textsf{QCMA})$. We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson-Kuperberg (CCC'07), Fefferman-Kimmel (MFCS'18)] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over $n$-bit boolean functions.