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 $m$ ($1\ensuremath{\le}m\ensuremath{\le}n$) 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 ${2}^{m} n$-locality inequality. Our findings provide a distinctive perspective on quantum correlation sharing in networks.