Summary: We first study a new family of graded quiver varieties together with a new \(t\)-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Y. Kimura and the author [Adv. Math. 262, 261–312 (2014; Zbl 1331.13016)]. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual Poincaré-Birkhoff-Witt basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras.