  Among $\mathbf{PSPACE}$-complete problems, QSAT, or quantified SAT, is one of
the most used to show that the class of problems solvable in polynomial time by
families of a given variant of P systems includes the whole $\mathbf{PSPACE}$.
However, most solutions require a membrane nesting depth that is linear with
respect to the number of variables of the QSAT instance under consideration.
While a system of a certain depth is needed, since depth 1 systems only allows
to solve problems in $\mathbf{P^{\#P}}$, it was until now unclear if a linear
depth was, in fact, necessary. Here we use P systems with active membranes with
charges, and we provide a construction that proves that QSAT can be solved with
a sublinear nesting depth of order $\frac{n}{\log n}$, where $n$ is the number
of variables in the quantified formula given as input.
