On $\#P\subseteq FP^{\Sigma_{f(n)}^P}$? Is it known that permanent of a $0/1$ $n\times n$ matrix $M$ is computable in polynomial or randomized polynomial time with access to a ${\Sigma_{(\log n)^c}^P}$ oracle where $0&lt;c$ holds and $\Sigma_{f(n)}^P$ means the number of alternations is $f(n)$?
If not known what is the consequence if it is possible to do so?
This is very unlikely to hold, but (as usual) impossible to rule out using current techniques, as we can’t even prove $\mathit{PSPACE}\ne P$.

However, we can at least show that no relativizing argument proves such an inclusion, i.e., there are oracles $X$ such that randomized polynomial-time with a $(\Sigma^P_{(\log n)^c})^X$ oracle does not include $\#P^X$. Better yet, there are oracles where it does not even include $\oplus P^X$ (which is included in $P^{\#P^X}$): otherwise, there would exist $O((\log\log n)^c)$-depth quasipolynomial-size unbounded fan-in circuits that approximate parity with error, say, $1/4$, contradicting Håstad’s theorem.