  In two-party, two-input and two-output measurement scenario only relevant
Bell's inequality is the Clauser-Horne-Shimony-Holt (CHSH) form. They also
provide the necessary and sufficient conditions for local realism. Any other
form, such as, Clauser-Horne and Wigner forms reduce to the CHSH one. Standard
Leggett-Garg inequalities are often considered to be the temporal analog of
CHSH inequalities. But, they do not provide the necessary and sufficient
conditions for macrorealism. There is thus scope of formulating new
macrorealist inequalities different and stronger than the standard Leggett-Garg
inequalities for testing compatibility between the macrorealism and quantum
theory. In this paper, we propose three different classes of macrorealistic
inequalities in three-time scenario; (i) The ones equivalent to standard
Leggett-Garg inequalities in both macrorealist model and in quantum theory.
(ii) A class of inequalities which are equivalent to the standard ones in
macrorealist model but inequivalent and stronger in quantum theory (iii)
Another class of inequalities which are inquivalent to the all formulations of
Leggett-Garg inequalities both in macrorealist model and in quantum theory.
This class of macrorealist inequalities reveals the incompatibility between
macrorealism and quantum theory for specific cases even when any other
formulation of Leggett-Garg inequalities fail to do so. We extend the
formulations of the class (ii) inequalities to the four-time and two-time
measurement scenario. Further, we provide a brief discussion about the
alternate formulation of macrorealism which was derived based on the
no-signaling in time conditions.
