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We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category   O   of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type   A  , we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type    A  1   , we identify remarkable relations in this quantum Grothendieck ring.

Quantum Grothendieck rings as quantum cluster algebras