Is properly quantified 3SAT complete for PSPACE and all PH levels?
QSAT with only 2 literals per clause was shown to be in P back in 1979.

Since the $\Sigma_k$ and $\Pi_k$ versions of QSAT can be reduced to equisatisfiable problems with 3 literals per clause using only a slight modification the method used to transform a SAT instance to 3SAT (i.e. putting all the new variables in the innermost existential group), those limited problems continue to be complete for their levels of the polynomial hierarchy.