Quantum complexity of TQBF
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