  We propose a scheme to characterize the non-Markovian dynamics and quantify
the non-Markovianity via the non-classicality measured by the negativity of
quantumness. By considering a qubit in contact with a critical Ising spin bath
and introducing an ancilla, we show that revivals of negativity of quantumness
indicate the non-Markovian dynamics. Furthermore, a normalized measure of
non-Markovianity based on the negativity of quantumness is introduced and the
influences of bath criticality, bath temperature and bath size on the
non-Markovianity are discussed. It is shown that, at the critical point, the
decay of non-Markovianity versus the size of spin bath is fastest and the
non-Markovianity is exactly zero only in the thermodynamic limit. Besides,
non-trivial behaviours of negativity of quantumness such as sudden change,
double sudden changes and keeping constant are found for different relations
between parameters of the initial state. Finally, how the non-classicality of
the system is affected by a series of bang-bang pulses is also examined.
