Conditions for macrorealism for systems described by many-valued variables

Macrorealism (MR) is the view that a system evolving in time possesses definite properties independent of past or future measurements and is traditionally tested for systems described at each time by a single dichotomic variable Q. A number of necessary and sufficient conditions for macrorealism have been derived for a dichotomic variable using sets of Leggett-Garg (LG) inequalities, or the stronger no signaling in time (NSIT) conditions, or a combination thereof. Here we extend this framework by establishing necessary and sufficient conditions for macrorealism for measurements made at two and three times for systems described by variables taking three or more values at each time. Our results include a generalization of Fine's theorem to many-valued variables for measurements at three pairs of times and we derive the corresponding complete set of LG inequalities. We find that LG inequalities and NSIT conditions for many-valued variables do not enjoy the simple hierarchical relationship exhibited by the dichotomic case. This sheds light on some recent experiments on three-level systems which exhibit a LG inequality violation even though certain NSIT conditions are satisfied. Under measurements of dichotomic variables using the Liiders projection rule the three-time LG inequalities cannot be violated beyond the Liiders bound (which coincides numerically with the Tsirelson bound obeyed by correlators in Bell experiments), but this bound can be violated in LG tests using degeneracy-breaking (von Neumann) measurements. We identify precisely which MR conditions are violated under these circumstances.