I know 3SAT is NP-complete and QSAT is PSPACE-complete. However, is it true that
$$\exists X_1 \forall X_2 \cdots Q_k X_k \colon \varphi(X_1, \ldots, X_k)$$
is complete for $\Sigma_k$, the existential side of level $k$ of the polynomial hierarchy if each clause in $\varphi$ has at most 3 literals (or some other constant size bound)? Similarly for
$$\forall X_1 \exists X_2 \cdots Q_k X_k \colon \varphi(X_1, \ldots, X_k)$$
in relation to $\Pi_k$, and the quantified version with unbounded alternation in relation to PSPACE, once again with each clause of $\varphi$ having at most 3 literals (or some other constant number)?