Probing various formulations of macrorealism for unsharp quantum measurements

Standard Leggett and Garg inequalities (SLGIs) were formulated for testing the incompatibility between the classical worldview of macrorealism and quantum mechanics. In recent times, various other formulations, such as Wigner form of LGIs (WLGIs), entropic LGIs (ELGIs) and the no-signaling in time (NSIT) condition have also been proposed. It is also recently argued that no set of SLGIs can provide the necessary and sufficient conditions for macrorealism but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulations of LGIs for testing macrorealism pertaining to the two different unsharp measurements. While the violations of WLGIs are more robust than SLGIs and ELGIs for spin-POVMs, here we demonstrate that for the case of biased POVMs, the quantum violations of both SLGIs and ELGIs provide the same robustness as WLGIs. Importantly, the violations of all formulations of LGIs can be achieved for \textit{any non-zero value} of unsharpness parameter. We have also studied the connection between LGIs and NSIT conditions. Further, we investigate the role of the joint measurability of the POVMs in the violation of LGIs and found that there is no generic connection.