Lower bound on alternations needed in $BQP$ versus $PH$ result?
If you just want oracle separations with $\#P$, you don't need to use the new result of Raz and Tal. You can use the classic Parity/Majority not in $AC^0$ results from the 1980s. 

For example, the strongest quantitative version of Parity not in $AC^0$ says that the $n$-bit Parity function cannot be computed by a quasi-polynomial size $AC^0$ circuit of depth $o(\log n/ \log\log n)$. So scaling this up, we get an oracle separation between $\oplus P$, which is in $\# P$, and $PH$ with $o(n/\log n)$ quantifiers.