Quantum violations of L\"uders bound Leggett-Garg inequalities for quantum channel

Leggett Garg inequalities (LGIs) provide an elegant way for probing the incompatibility between the notion of macrorealism and quantum mechanics. For unitary dynamics, the optimal quantum violation of a LGI is constrained by the L\"uders bound. In this paper, we have studied two formulations of LGIs in a three-time LG scenarios, viz., the standard LGIs and third-order LGIs both for unbiased and biased measurement settings. We show that if the system evolves under a non-unitary quantum channel between two measurements, the quantum violations of both forms of LGIs exceed their respective L\"uders bounds and can even reach their algebraic maximum in sharp measurement settings. We found that when the measurement is unsharp the quantum violations of both standard and third-order LGIs for non-unitary quantum channels can be obtained for the lower value of the unsharpness parameter compared to the unitary dynamics. We critically examined the violation of L\"uders bound of LGIs and its relation to the violation of various no-signaling in time conditions, another formalism for testing macrorealism. It is shown that mere violations of no-signaling conditions are not enough to warrant the violation of standard LGIs, an interplay between the violations of various NSIT condition along with a threshold value play an important role. On the other hand violation of third-order LGI is obtained when the degree of violation of a specific no-signaling in time condition reaches a different threshold value.