Diagonalization in oracle separation between QMA and PP


Reposting from cstheorystackexchange for more visibility:

Diagonalization is a very common technique to find oracle separations. For example, it can be used to separate $ \cal{P}$ and $ \cal{NP}$ , with the essential idea being that of constructing an oracle in stages and diagonalizing any $ \cal{P}$ machine against that oracle. Similarly, diagonalization arguments can also be used to diagonalize a $ \cal{BQP}$ machine against an oracle like the Grover oracle, and achieve a separation between $ \cal{BQP}$ and $ \cal{QMA}$ . I was wondering whether I can use diagonalization (against $ \cal{QMA}$ machines) to separate classes like $ \cal{QMA}$ and $ \cal{PP}$ , or $ \cal{QMA}$ and $ \cal{AWPP}$ . Is there any literature on these types of separations? A subtlety that I note is that for the diagonalization argument against $ \cal{BQP}$ machines, the essential idea is that the quantum machine cannot “search for a needle in a haystack”, meaning that if there are an exponential number of query states to keep track of, a quantum machine is almost blind to the change in any one of them. However, if you have a prover as well as a quantum machine, the prover can just “give you the needle”; meaning, the prover can just send you the right state to query. Can diagonalization still work in these settings?

As one of the answers suggested, a way to approach an oracle separation between $ \cal{QMA}$ and $ \cal{PP}$ is to consider a language $ L$ , for any language $ A$ , such that $ $ L = \{1^{n} : |A \cap \{0, 1\}^{n}| > \frac{1}{2} 2^{n} \}.$ $

Clearly, $ L \in \cal{PP}^{A}$ , and it is reasonable to think that $ L \notin \cal{QMA}^{A}$ and we can use diagonalization to show it. However, for me, the analysis of the latter is getting complicated as I do not how to mathematically model the prover’s action: the prover can send any quantum state and can potentially send a quantum state unrelated to the previous one when the oracle is changed during diagonalization. I am looking for any reference or any explicit proof of this non-containment.