Network nonlocality sharing in the star-shaped scenario using only projective measurements

Sequential measurements present a novel framework for constructing large-scale quantum networks, particularly in quantum correlation sharing. In this paper, we design projective measurements and combination strategies to achieve nonlocality sharing in star-shaped networks by leveraging shared classical randomness. This network consists of n branches, each containing one or two sequential observers. By investigating two-sided nonlocality sharing within a 2-local network, we identify the optimal projective measurement combination strategy based on the double violation of the 2-locality inequality. Utilizing this optimal strategy, we demonstrate that four inequalities can be violated simultaneously in the 2-local scenario. Furthermore, in an n-local scenario involving m (1≤m≤n) branches with two sequential observers, our optimal strategy shows that non-n-locality sharing among all observers can be achieved through the simultaneous violation of 2m n-locality inequality. Our findings provide a distinctive perspective on quantum correlation sharing in networks.