Recently, Xu et al. [Phys. Rev. A 111, 022416 (2025)] have constructed minimal nonlocal orthogonal product states (OPSs) on $n$-partite systems ($n\ge 3$), as well as minimal nonlocal orthogonal entangled states (OESs) on both bipartite and $n$-partite systems ($n\ge 3$). A meaningful question is whether entanglement-assisted discrimination protocols with less entanglement than teleportation can be designed for the above OPSs and OESs. This paper provides an affirmative answer to this question and diagrammatically illustrates the entanglement-assisted local discrimination protocols through geometric shapes. First, we present two efficient entanglement-assisted local discrimination protocols for nonlocal OPSs on tripartite systems, and extend both protocols to general $n$-partite systems ($n\ge 3$). Furthermore, we implement the efficient local discrimination of nonlocal OESs on both bipartite and tripartite systems by using auxiliary entanglement, and the discrimination method is generalized to $n$-partite systems ($n\ge 3$). In particular, when distinguishing tripartite nonlocal OESs, we construct local discrimination protocols with less entanglement resource for $d=4$ and $d=5$. Our results highlight the essential role of entanglement in identifying minimal nonlocal sets on multipartite systems, revealing the phenomenon of less nonlocality with more entanglement.

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• Quantum information processing
• Quantum information theory