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  1. 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 $0<c$ holds and $\Sigma_{f(n)}^P$ means the number of alternations is $f(n)$?

  2. If not known what is the consequence if it is possible to do so?

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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.

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