This paper explores the realization of non-Abelian fractional quantum anomalous Hall (FQAH) states in topological minibands of moiré systems, specifically in skyrmion Chern band (SCB) models. The study focuses on the possibility of realizing non-Abelian phases, such as the Moore-Read state, in the second miniband of twisted transition metal dichalcogenide (TMD) homobilayers. Using many-body exact diagonalization, the authors demonstrate that the Moore-Read state can be realized at half-filling of the second miniband, despite strong variations in Berry curvature in momentum space. This finding highlights the feasibility of non-Abelian fractionalization in moiré systems without Landau levels and provides insights into the conditions necessary for their realization. The research builds on recent experimental observations of FQAH states in twisted bilayer semiconductors and rhombohedral pentalayer graphene/hBN at zero magnetic field. Theoretical predictions suggest that these states arise from Coulomb interactions in partially filled topological moiré bands and spontaneous time-reversal symmetry breaking. The study also discusses the potential for non-Abelian phases in other moiré material platforms, including twisted bilayer graphene and narrow gap semiconductors. The paper introduces a family of SCB models, which capture the essential features of topological minibands in twisted TMD bilayers. These models are shown to host non-Abelian FQAH states, with the Moore-Read state being a candidate. The study uses a Hamiltonian that describes electrons coupled to a skyrmion-like spin or pseudospin texture, leading to flat topological minibands. The effective Hamiltonian for the system is derived, and the results show that the non-Abelian state can be realized in the second miniband. The research also addresses the question of why SCB models host non-Abelian FQAH states resembling those of the half-filled first Landau level. The study quantifies the similarity between the SCB miniband and the Landau level using the "LL weight," which is found to be high in the SCB models. This indicates that the SCB models can support non-Abelian fractionalization without the need for a magnetic field. The paper concludes that the realization of non-Abelian topological order in moiré systems is feasible and offers significant potential for high-temperature topological protection. The study provides a realistic material proposal for realizing non-Abelian phases in topological minibands and highlights the desirable conditions for their realization. The findings contribute to the broader understanding of strongly correlated topological quantum matter and open new avenues for exploring non-Abelian fractionalization in moiré systems.This paper explores the realization of non-Abelian fractional quantum anomalous Hall (FQAH) states in topological minibands of moiré systems, specifically in skyrmion Chern band (SCB) models. The study focuses on the possibility of realizing non-Abelian phases, such as the Moore-Read state, in the second miniband of twisted transition metal dichalcogenide (TMD) homobilayers. Using many-body exact diagonalization, the authors demonstrate that the Moore-Read state can be realized at half-filling of the second miniband, despite strong variations in Berry curvature in momentum space. This finding highlights the feasibility of non-Abelian fractionalization in moiré systems without Landau levels and provides insights into the conditions necessary for their realization. The research builds on recent experimental observations of FQAH states in twisted bilayer semiconductors and rhombohedral pentalayer graphene/hBN at zero magnetic field. Theoretical predictions suggest that these states arise from Coulomb interactions in partially filled topological moiré bands and spontaneous time-reversal symmetry breaking. The study also discusses the potential for non-Abelian phases in other moiré material platforms, including twisted bilayer graphene and narrow gap semiconductors. The paper introduces a family of SCB models, which capture the essential features of topological minibands in twisted TMD bilayers. These models are shown to host non-Abelian FQAH states, with the Moore-Read state being a candidate. The study uses a Hamiltonian that describes electrons coupled to a skyrmion-like spin or pseudospin texture, leading to flat topological minibands. The effective Hamiltonian for the system is derived, and the results show that the non-Abelian state can be realized in the second miniband. The research also addresses the question of why SCB models host non-Abelian FQAH states resembling those of the half-filled first Landau level. The study quantifies the similarity between the SCB miniband and the Landau level using the "LL weight," which is found to be high in the SCB models. This indicates that the SCB models can support non-Abelian fractionalization without the need for a magnetic field. The paper concludes that the realization of non-Abelian topological order in moiré systems is feasible and offers significant potential for high-temperature topological protection. The study provides a realistic material proposal for realizing non-Abelian phases in topological minibands and highlights the desirable conditions for their realization. The findings contribute to the broader understanding of strongly correlated topological quantum matter and open new avenues for exploring non-Abelian fractionalization in moiré systems.