Is there a version of Toda's theorem for intermediate levels of the polynomial hierarchy ? More precisely, is there any variant of the Toda's theorem that states:

Let $\# wSAT$ be the number of satisfying assignments to a boolean formula, where $(\log_2 n)^w$ variables (out of $n$) are assigned to TRUE.

$\bigcup_{i=1}^k \Sigma^P_k \subseteq P^{\# wSAT}$ ?

If something of this form is known, a reference would greatly help.


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