A toric variety is a normal complex variety which is completely described by combinatorial data, namely by a fan of strongly convex rational (with respect to a lattice) cones. Due to this rationality condition, toric varieties are (equivariantly) rigid since a deformation of the lattice can make it dense. A solution to this problem consists in considering quantum toric stacks. The latter is a stacky generalization of toric varieties where the "lattice" is replaced by a finitely generated subgroup of $\mathbb{R}^d$ (in the simplicial case as introduced by L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky). The goal of this paper is to explain the moduli spaces of quantum toric stacks and their compactification.