  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.
