In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in $m_1otimes m_2otimesldotsotimes m_n$ , where the quantum system $m_1$ belongs to Alice, $m_2$ belongs to Bob, ldots and $m_n$ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in $m_1otimes m_2otimes m_3$. With the conclusion in $m_1otimes m_2otimes m_3$, we prove distinguishability or indistinguishability of some quantum states. At last, we give the $n$-party distinguishable quantum states in $m_1otimes m_2otimescdotsotimes m_n$. Our study further reveals quantum nonlocality in multipartite high-dimensional.
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