  We introduce a monoidal analogue of Jantzen filtrations in the framework of
monoidal abelian categories with generic braidings. It leads to a deformation
of the multiplication of the Grothendieck ring. We conjecture, and we prove in
many remarkable situations, that this deformation is associative so that our
construction yields a quantization of the Grothendieck ring as well as analogs
of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional
representations of simply-laced quantum loop algebras, we prove the
associativity and we establish that the resulting quantization coincides with
the quantum Grothendieck ring constructed by Nakajima and Varagnolo-Vasserot in
a geometric manner. Hence, it yields a unified representation-theoretic
interpretation of the quantum Grothendieck ring. As a second main example, we
establish an analogous result for a monoidal category of finite-dimensional
modules over symmetric quiver Hecke algebras categorifying the coordinate ring
of a unipotent group associated with a Weyl group element.
