Comparing quantum Markovianities: Distinguishability versus correlations

We compare several criteria for quantum Markovianities and illustrate the differences and similarities through explicit examples of random unitary qubit dynamics (Pauli channels). In particular, we demonstrate that the quantum Markovianity based on distinguishability [Breuer et al., Phys. Rev. Lett. 103, 210401 (2009)] is not equivalent to that based on correlations [Luo et al., Phys. Rev. A 86, 044101 (2012)].