Strongest quantum nonlocality in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-partite systems

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 ${({\mathbb{C}}^{3})}^{\ensuremath{\bigotimes}N}$. 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 ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$ for $d\ensuremath{\ge}4$, 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.