Quantum cluster algebras and representations of shifted quantum affine algebras

We construct a new quantization Kt(O_Z) of the Grothendieck ring of the category O_Z of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring Kt(O^b_Z) for the quantum Borel affine algebra, namely that there is a natural embedding Kt(O^b_Z)↪Kt(O_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 A1). 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 ℂt[SL^{w0,w0}]