Experimental realizations of Abelian fractional Chern insulators have demonstrated the potentials of moiré systems in synthesizing exotic quantum phases. Remarkably, a twisted multilayer graphene system may also host non-Abelian states competing with charge density waves under Coulomb interactions. Here, through larger-scale exact diagonalization simulations, we map out the quantum phase diagram for a $\nu =1/2$ system with electrons occupying the lower moiré band of a double twisted bilayer graphene. By increasing the system size, we find the ground state has sixfold near degeneracy and with a finite spectral gap separating the ground states from the excited states across a broad range of parameters. Further computation of many-body Chern numbers establish the topological order of the state, and we rule out the possibility of charge density wave orders based on a featureless density structure factor. Furthermore, we inspect the particle-cut entanglement spectrum to identify the topological state as a non-Abelian Moore-Read state. Combining all the above evidence, we conclude that the Moore-Read ground state dominates the quantum phase diagram for a double twisted bilayer graphene system for a broad range of coupling strengths with realistic Coulomb interactions.

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• Anyons
• Fractional quantum Hall effect
• Topological phases of matter
• Twisted bilayer graphene
• Exact diagonalization