  We give a definition of monoidal categorifications of quantum cluster
algebras and provide a criterion for a monoidal category of finite-dimensional
graded $R$-modules to become a monoidal categorification of a quantum cluster
algebra, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly
speaking, this criterion asserts that a quantum monoidal seed can be mutated
successively in all the directions once the first-step mutations are possible.
In the course of the study, we also give a proof of a conjecture of Leclerc on
the product of upper global basis elements.

Representation Theory;Quantum Algebra