Bell inequality, steering, incompatibility and Leggett–Garg inequality under coarsening measurement

We investigate the effects of the coarsening measurement on the quantum-to-classical transition by Bell–Clauser–Horne–Shimony–Holt (Bell–CHSH) non-locality for the conventional two-qubit system, the Leggett–Garg inequality for a two-level system, and steering and incompatibility both in the equatorial plane for N measurement settings. We find that for any fixed N, steering is more vulnerable than incompatibility for coarsening measurement both in reference and in final resolution. For \(N=2\) measurement settings, under the coarsening measurement reference the Leggett–Garg inequality is the most robust, Bell–CHSH non-locality lies between steering and incompatibility, while in the coarsening measurement of final resolution for \(N=2\) measurement settings incompatibility is the most robust, steering and Bell–CHSH non-locality are equally vulnerable, and more than the Leggett–Garg inequality. However, as N increases, incompatibility and steering will become more robust than the Leggett–Garg inequality under the coarsening measurement in reference and in final resolution, respectively. Finally, for the Leggett–Garg inequality, we find that the robustness of the coarsening measurement reference is more than the coarsening temporal reference. In one word, the effects of coarsening measurement strongly depend on the ways of coarsening.