Motivated by topology, we develop a general theory of traces and shadows in an endobicategory, which is a bicategory C with an endobifunctor \Sigma\colon \mathbf C \to\mathbf C . Applying this framework to the bicategory of Chen-Khovanov bimodules with identity as Σ we reproduce Asaeda-Przytycki-Sikora (APS) homology for links in a thickened annulus. If Σ is a reflection, we obtain the APS homology for links in a thickened M\"obius band. Both constructions can be deformed by replacing Σ with an endofunctor Σq such that \Sigma_q \alpha:=q^{-|\alpha|}\Sigma\alpha for any 2-morphism α and identity otherwise, where q is a fixed invertible scalar. We call the resulting invariant the \emph{quantum link homology}. We prove in the annular case that this homology carries an action of \mathcal U_q(\mathfrak{sl}_2), which intertwines the action of cobordisms. In particular, the quantum annular homology of an n-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups does depend on the quantum parameter q. Hence, our quantum link homology has a richer structure.
