For a quantified Boolean formula (QBF), the problem of computing the number of winning strategies is known as the #QBF problem. This problem is considered harder than the analogous #SAT problem. Recently, important proof systems for QBFs and #SAT have been studied. By extending the ideas from both fields, we show that it is possible to design proof systems for #QBF. Such proof systems are important not only for advancing the theory of #QBF but also for certifying and designing better #QBF solvers, an area that is still in its early stages.

In this paper, we explore #QBF proof systems to count the number of Skolem functions. In addition to a naive system, we study #QBF systems based on the $\forall$-expansion rule of QBFs. We observe that these systems have inherent structural weaknesses that lead to lower bounds. As an alternative, we propose a #QBF proof system that we call Q-MICE, which consists of sound inference rules for computing and certifying the #QBF solution, similar to the line-based #SAT proof system MICE. To demonstrate the strength of Q-MICE, we present various upper bounds, such as the quantified version of the propositional XOR-PAIRS formula, which is known to be hard for MICE. Consequently, we also separate Q-MICE from $\forall$-expansion based #QBF proof systems.