The $d$-th order Laplacian spectral moment of a $k$-uniform hypergraph is the sum of the $d$-th powers of all eigenvalues of its Laplacian tensor. In this paper, we obtain some expressions of the Laplacian spectral moments for $k$-uniform power hypergraphs, and these expressions can be represented by some parameters of graphs. And we show that some graphs can be determined by their high-order Laplacian spectrum by using the Laplacian spectral moments of power hypergraphs.