Moiré materials have emerged in the last years as a highly tunable and experimentally accessible platform to study strongly correlated topological phases of matter. In particular, twisted bilayers of graphene and transition metal dichalcogenides have been shown to be Chern insulators, where electrons break time-reversal symmetry and spontaneously polarize into a single spin and valley, occupying a band with non-zero Chern number. While Chern insulators are already interesting because they exhibit the integer quantum Hall effect in the absence of an external magnetic field, recent theoretical predictions and experimental realizations of fractional Chern insulators (FCIs) have opened up the possibility of exploring even more exotic states. These include phases hosting non-Abelian excitations, which are interesting for fault-tolerant topological quantum computing, states with Chern number higher than one that have no analog in the standard quantum Hall paradigm, and quantum Hall crystals exhibiting topological order with broken translation symmetry. In this project we will use memory-expensive numerical methods to investigate the competing topological quantum phases in moiré materials. After setting up the many-body Hamiltonian projected into the relevant moiré flat bands, we will employ exact diagonalization to obtain the energetically lowest eigenstates of the system. The initial identification of the system’s phase will be based on the ground state degeneracy and the respective (momentum) quantum numbers. A more complete analysis will include the calculation of particle entanglement spectra, which contain information about the allowed number of quasi-hole excitations that is characteristically unique to the specific phase. In addition, the structure factor and pair-correlation function can be computed to explore the liquid or crystal character of the ground state. Using these methods, we aim to shed light on the stability and nature of competing phases such as charge density waves, (composite) Fermi liquids, and Abelian as well as non-Abelian FCIs in moiré flat bands with Chern number equal to or larger than one. We expect our results to provide important insights on fundamental topics of condensed matter quantum many-body physics with potential impact in the future realization of topologically-protected quantum devices.