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Nonclassicality of temporal correlations pertaining to noncommutativesequential measurements is defined through the violation of macrorealisticinequalities, known as Leggett-Garg inequalities (LGI). We investigate theenergy cost of the process associated with the Leggett-Garg test in the contextof noiseless and Markovian noise for arbitrary initial states. We prove that innoiseless and in certain noisy scenarios, the maximal violations of LGI underthe energy constraint occurs when the average energy of the process is equal tothe negative of the energy of the initial state. Such a dependence of LGI onthe choice of the initial state is not seen in the unconstrained case.Moreover, we find that in the presence of a moderate amount of Markovian noise,the amount of violation of LGI remains almost unaltered with a suitable choiceof the evolution and dephasing operators in the neighborhood of the maximalviolation line, thereby showing the robustness of temporal correlations underenvironmental effects.

Canonical Leggett-Garg Inequality: Nonclassicality of temporal quantum correlations under energy constraint