  Quantum non-demolition measurements define a non-invasive protocol to extract
information from a quantum system that we aim to monitor. They exploit an
additional quantum system that is sequentially coupled to the system.
Eventually, by measuring the additional system, we can extract information
about temporal correlations developed by the quantum system dynamics with
respect to a given observable. This protocol leads to a quasi-probability
distribution for the measured observable outcomes, which can be negative. We
prove that the presence of these negative regions is a necessary and sufficient
condition for the violation of macrorealism. This is a much stronger condition
than the violation of the Leggett-Garg inequalities commonly used for the same
task. Indeed, we show that there are situations in which Leggett-Garg
inequalities are satisfied even if the macrorealism condition is violated. As a
consequence, the quantum non-demolition protocol is a privileged tool to
identify with certainty the quantum behavior of a system. As such, it has a
vast number of applications to different fields from the certification of
quantumness to the study of the quantum-to-classical transition.
