The strongest quantum nonlocality is a stronger form of nonlocality than that manifested by locally indistinguishable quantum states. It has efficient applications in quantum information hiding and quantum secret sharing. Constructing a smaller-size set with the strongest nonlocality contributes to reducing resource consumption. In this paper we establish a sufficient and necessary condition for the strongest nonlocality in general $N$-partite systems. Then we derive the minimum size of the strongest nonlocal set with symmetry in the system ${\left({\mathbb{C}}^3\right)}^{\otimes N}$. Notably, the genuinely entangled sets constructed in our previous work [Phys. Rev. A 109, 022220 (2024)] achieved this minimum size. In addition, we construct the strongest nonlocal genuinely entangled sets in the system ${\left({\mathbb{C}}^d\right)}^{\otimes N}$ for $d\ge 4$, which have a smaller size than the existing sets as $N$ increases. Our results provide a meaningful complement to the current understanding of the strongest nonlocality.

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• Quantum correlations, foundations & formalism
• Quantum foundations
• Quantum information processing
• Quantum information theory