<p>In this paper, we generalize the major results from Jing-Yang-Zhao’s paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established a correspondence between local unitary (LU) equivalence and simultaneous orthogonal (SO) equivalence of bipartite quantum states, and then used this correspondence to show that the problem of determining LU equivalence of bipartite states reduces to checking trace identities. In particular, we extend both results to tripartite quantum states. We are able to do this by utilizing a hypermatrix algebra framework and by applying a powerful generalization of Specht’s criterion proved in Futorny-Horn-Sergeichuk’s paper "Specht’s Criterion for Systems of Linear Mappings." With our established hypermatrix algebra framework and the aforementioned generalization of Specht’s criterion, it is apparent that our results can be extended to arbitrary multipartite quantum states, however there are some practical limitations which are explored and discussed towards the end.</p>