Sharing quantum nonlocality and genuine nonlocality with independent observables

Recently the authors of Phys. Rev. Lett. 125, 090401 (2020) considered following scenario: Alice and Bob each have half a pair entangled qubit states. measures his then passes part to second who again, so on. The goal is maximize number Bobs that can an expected violation Clauser-Horne-Shimony-Holt (CHSH) inequality with single Alice. By taking maximally pure two-qubit state $|\ensuremath{\phi}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})$ as example, it has been constructively proved arbitrarily many independent share nonlocality Here we demonstrate observers arbitrary dimensional bipartite but not necessary state. Furthermore, generalized Greenberger-Horne-Zeilinger (GHZ) states show at most two Charlies genuine GHZ Bob.