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The local and nonlocal Allen-Cahn equations (ACEs) have received increasing attention in the study of the complicated interfacial problems. In this paper, we conduct a comparison between local and nonlocal ACEs with the property of mass conservation in the framework of lattice Boltzmann (LB) method. To this end, we first propose two simple multiple-relaxation-time LB models for local and nonlocal ACEs, and through the Chapman-Enskog expansion, the local and nonlocal ACEs can be recovered correctly from the developed LB models. Then we test these two LB models with several examples, and the numerical results show that the developed LB models are accurate and also have a second-order convergence rate in space. Finally, a comparison between the local and nonlocal ACEs is also performed in terms of mass conservation, stability and accuracy. The results show that both local and nonlocal ACEs …