Prove that QBF, defined below, belongs to PSPACE.
Input: a quantified Boolean formula $$F = (Q_1x_1)(Q_2x_2) · · · (Q_nx_n)B(x_1, . . . , x_n)$$ where $B(x_1, . . . , x_n)$ is a Boolean expression in the variables $x_1, . . . , x_n$ and each $Q_i$ is a quantifier $\forall$ or $\exists$.
Question: is F true?
This is an exercise from chapter three of Complexity and cryptography by John Talbot and Dominic Welsh. Unfortunately I'm unsure how one must tackle such problems which are about PSPACE. I know the SAT problem is NP-COMPLETE and F is quite similar to it. (or is it not?) Please guide me trough the steps of its proof. I also know $NP \subset PSPACE$, so if we prove that this is NP it's PSPACE as well.