  There are strong evidences in the literature that quantum non-Markovianity
would hinder the presence of Quantum Darwinism. In this Letter, we study the
relation between quantum Darwinism and approximate quantum Markovianity for
open quantum systems by exploiting the properties of quantum conditional mutual
information. We show that for approximately Markovian quantum processes the
conditional mutual information still has the scaling property for Quantum
Darwinism. Then two general bounds on the backflow of information are obtained,
with which we can show that the presence of Quantum Darwinism restricts the
information backflow and the quantum non-Markovianity must be small.
