  Quantum channels breaking entanglement, incompatibility, or nonlocality are
defined as such because they are not useful for entanglement-based, one-sided
device-independent, or device-independent quantum information processing,
respectively. Here, we show that such breaking channels are related to
complementary tests of macrorealism i.e., temporal separability, channel
unsteerability, temporal unsteerability, and the temporal Bell inequality. To
demonstrate this we first define a steerability-breaking channel, which is
conceptually similar to entanglement and nonlocality-breaking channels and
prove that it is identical to an incompatibility-breaking channel. A hierarchy
of quantum non-breaking channels is derived, akin to the existing hierarchy
relations for temporal and spatial quantum correlations. We then introduce the
concept of channels that break temporal correlations, explain how they are
related to the standard breaking channels, and prove the following results: (1)
A robustness-based measure for non-entanglement-breaking channels can be probed
by temporal nonseparability. (2) A non-steerability-breaking channel can be
quantified by channel steering. (3) Temporal steerability and non-macrorealism
can be used for, respectively, distinguishing unital steerability-breaking
channels and nonlocality-breaking channels for a maximally entangled state.
Finally, a two-dimensional depolarizing channel is experimentally implemented
as a proof-of-principle example to demonstrate the hierarchy relation of
non-breaking channels using temporal quantum correlations
