Determine whether multiple rotated parties can improve GMNL detection

Determine whether introducing multiple rotated parties can improve the ratio between the maximal quantum value and the bi-local bound for genuine multipartite nonlocality inequalities on graph states with graph structures beyond the examples analyzed in the paper.

Background

The paper investigates Bell inequalities for detecting genuine multipartite nonlocality in graph states, including variants of the BASTA construction with more than one rotated party. Although the authors exhibit a four-party line-graph example in which adding a second rotated party prevents GMNL detection, they do not establish that this behavior holds for all graph structures.

The unresolved issue is whether some other graph states may benefit from additional rotated sites, yielding a larger ratio between the quantum violation and the bi-local bound and thereby improving noise robustness.

References

Even though we have provided an example for which such a construction fails in the sense that an additional rotation does not increase the ratio $\beta_Q/\beta_{C}$, we do not have a proof that introducing more rotated sites does not offer benefits for some other graph structures. This aspect certainly requires further investigation.

— Genuine multipartite nonlocality detection in graph states  (2610.13609 - Makuta et al., 9 Oct 2026) in Section Conclusions