  We consider the quantum cluster algebras which are injective-reachable and
introduce a triangular basis in every seed. We prove that, under some initial
conditions, there exists a unique common triangular basis with respect to all
seeds. This basis is parametrized by tropical points as expected in the
Fock-Goncharov conjecture.
  As an application, we prove the existence of the common triangular bases for
the quantum cluster algebras arising from representations of quantum affine
algebras and partially for those arising from quantum unipotent subgroups. This
result implies monoidal categorification conjectures of Hernandez-Leclerc and
Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond
to simple modules.
