  We extend the notion of the quantization of the coefficients of the ordinary
cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In
parallel to the ordinary case, it is tightly integrated with certain
generalizations of the ordinary quantum dilogarithm, which we call the quantum
dilogarithms of higher degrees. As an application, we derive the identities of
these generalized quantum dilogarithms associated with any period of quantum
Y-seeds.
