Quantum complexity of TQBF There is no classical algorithm for $n$-bit TQBF with better than $O(2^n)$ complexity.  Is that also the best known bound for quantum algorithms / circuits?
Edit: As pointed out by Huck Bennett, in the full alternation subset of TQBF we can in fact beat $O(2^n)$ via randomized tree search.  This is the case I care about most, so I would also be curious for the best exponent for a quantum algorithm if there are all $n-1$ quantifier flips.
The best quantum algorithm for QBF on n variable formulas of size s runs in about $2^{n/2}poly(s)$ steps, regardless of quantifier alternations. This is due to a series of advances on quantum algorithms for game tree search in the early 2000s.
Here is the first reference I found from googling; there are more: https://arxiv.org/abs/0907.1623