<p>In this paper, we propose a local discontinuous Galerkin (LDG) method for the square phase field crystal equation that contains much higher degree of nonlinearity. The energy stability and optimal error estimates of the semi-discrete LDG method are rigorously proved. To avoid the severe time step restriction of explicit time marching methods, we construct and analyze energy stable time discretizations which allow much larger time steps. To this end, we add an additional term consistent with the time discretization order to stabilize the numerical schemes. Then, we rigorously prove the energy stability of the corresponding schemes under the LDG framework. Numerical experiments demonstrate the accuracy, energy stability and capability of the proposed methods for solving the square phase field crystal equation.</p>