Motivated by the recent work of QED$_3$-Chern-Simons quantum critical points of fractional Chern insulators (Phys. Rev. X textbf{8}, 031015, (2018)), we study its non-Abelian generalizations, namely QCD$_3$-Chern-Simons quantum phase transitions of fractional Chern insulators. These phase transitions are described by Dirac fermions interacting with non-Abelian Chern-Simons gauge fields ($U(N)$, $SU(N)$, $USp(N)$, etc.). Utilizing the level-rank duality of Chern-Simons gauge theory and non-Abelian parton constructions, we discuss two types of QCD$_3$ quantum phase transitions. The first type happens between two Abelian states in different Jain sequences, as opposed to the QED3 transitions between Abelian states in the same Jain sequence. A good example is the transition between $sigma^{xy}=1/3$ state and $sigma^{xy}=-1$ state, which has $N_f=2$ Dirac fermions interacting with a $U(2)$ Chern-Simons gauge field. The second type is naturally involving non-Abelian states. For the sake of experimental feasibility, we focus on transitions of Pfaffian-like states, including the Moore-Read Pfaffian, anti-Pfaffian, particle-hole Pfaffian, etc. These quantum phase transitions could be realized in experimental systems such as fractional Chern insulators in graphene heterostructures.
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