  We establish ring isomorphisms between quantum Grothendieck rings of certain
remarkable monoidal categories of finite-dimensional representations of quantum
affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in
part on the corresponding quantum cluster algebra structures. Moreover, we
prove that our isomorphisms specialize at $t = 1$ to the isomorphisms of
(classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by
other methods. As a consequence, we prove a conjecture formulated by the first
author in 2002 : the multiplicities of simple modules in standard modules in
the categories above for type $B_n^{(1)}$ are given by the specialization of
certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these
polynomials are positive.
