  In this paper, we introduce a quantum version of the wonderful
compactification of a group as a certain noncommutative projective scheme. Our
approach stems from the fact that the wonderful compactification encodes the
asymptotics of matrix coefficients, and from its realization as a GIT quotient
of the Vinberg semigroup. In order to define the wonderful compactification for
a quantum group, we adopt a generalized formalism of $\mathsf{Proj}$ categories
in the spirit of Artin and Zhang. Key to our construction is a quantum version
of the Vinberg semigroup, which we define as a $q$-deformation of a certain
Rees algebra, compatible with a standard Poisson structure. Furthermore, we
discuss quantum analogues of the stratification of the wonderful
compactification by orbits for a certain group action, and provide explicit
computations in the case of $\mathrm{SL}_2$.

Representation Theory;Algebraic Geometry;Quantum Algebra