Bounded-width QBF is PSPACE-complete

•We solve an open problem on the complexity of bounded tree-width QBF.•We show that the bounded tree-width QBF problem is PSPACE-complete.•We present a family of formulas with bounded tree-width with short refutations.

Tree-width and path-width are two well-studied parameters of structures that measure their similarity to a tree and a path, respectively. We show that QBF on instances with constant path-width, and hence constant tree-width, remains PSPACE-complete. This answers a question by Vardi. We also show that on instances with constant path-width and a very slow-growing number of quantifier alternations (roughly inverse-Ackermann many in the number of variables), the problem remains NP-hard. Additionally, we introduce a family of formulas with bounded tree-width that do have short refutations in Q-resolution, the natural generalization of resolution for quantified Boolean formulas.