Can $\mathsf{P}^{\#\mathsf{P}}$ be described in terms of a non-deterministic (alternating) Turing machine?

Can the $$\mathsf{P}^{\#\mathsf{P}}$$ (= $$\mathsf{P}^{\mathsf{PP}}$$) class be described in terms of a non-deterministic Turing machine (in particular, an alternating Turing machine)? And would a $$\mathsf{P}^{\#\mathsf{P}}$$ machine be equivalent in computational power to any non-deterministic Turing machine? I suppose it should be because $$\mathsf{P}^{\#\mathsf{P}}$$ lies in between $$\mathsf{PSPACE}$$ and $$\mathsf{PH}$$ as per this, and both $$\mathsf{PSPACE}$$ and $$\mathsf{PH}$$ are describable in terms of alternating Turing machines. However, I could not find any description of $$\mathsf{P}^{\#\mathsf{P}}$$ or $$\mathsf{P}^{\mathsf{PP}}$$ in terms of alternating Turing machines, so far.

• It is an open problem weather under SETH, there's a tight reduction from QBF-SAT TO #CNF-SAT as per this talk (~2 minutes before the ending): youtube.com/watch?v=CIsfsGGOmKI Dec 22 '19 at 2:32
• @S.D. You could define "Threshold" Turing Machines using counting quantifiers instead of alternating quantifiers and they capture the Counting Hierarchy and in particular $P^{\# P}$. See for example, researchgate.net/profile/Eric_Allender/publication/… Dec 22 '19 at 14:05