We construct a new quantization \(K_t(\mathcal {O}^{\text {sh}}_{\mathbb {Z}})\) of the Grothendieck ring of the category \(\mathcal {O}^{\text {sh}}_{\mathbb {Z}}\) of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring \(K_t(\mathcal {O}^{\mathfrak {b},+}_{\mathbb {Z}})\) for the quantum Borel affine algebra, namely that there is a natural embedding \(K_t(\mathcal {O}^{\mathfrak {b},+}_{\mathbb {Z}})\hookrightarrow K_t(\mathcal {O}^{\text {sh}}_{\mathbb {Z}})\) . Our construction is partially based on the cluster algebra structure on the classical Grothendieck ring discovered by Geiss–Hernandez–Leclerc. As first applications, we formulate a quantum analogue of QQ-systems (that we make completely explicit in type \(A_1\) ). We also prove that the quantum oscillator algebra is isomorphic to a localization of a subalgebra of our quantum Grothendieck ring and that it is also isomorphic to the Berenstein–Zelevinsky’s quantum double Bruhat cell \(\mathbb {C}_t[{{\,\textrm{SL}\,}}_2^{w_0,w_0}]\) .
