  Quantum non-Markovianity in tripartite quantum states $\rho_{ABC}$ represents
a correlation between systems $A$ and $C$ when conditioned on the system $B$
and is known to have both classical and quantum contributions. However, a
systematic characterization of the latter is missing. To address this, we
propose a faithful measure for non-Markovianity of genuine quantum origin
called squashed quantum non-Markovianity (sQNM). It is based on the quantum
conditional mutual information and is defined by the left-over non-Markovianity
after squashing out all non-quantum contributions. It is lower bounded by the
squashed entanglement between non-conditioning systems in the reduced state and
is delimited by the extendibility of either of the non-conditioning systems. We
show that the sQNM is monogamous, asymptotically continuous, convex, additive
on tensor-product states, and generally super-additive. We characterize genuine
quantum non-Markovianity as a resource via a convex resource theory after
identifying free states with vanishing sQNM and free operations that do not
increase sQNM in states. We use our resource-theoretic framework to bound the
rate of state transformations under free operations and to study state
transformation under non-free operations; in particular, we find the quantum
communication cost from Bob ($B$) to Alice ($A$) or Charlie ($C$) is lower
bounded by the change in sQNM in the states. The sQNM finds operational
meaning; in particular, the optimal rate of private communication in a variant
of conditional one-time pad protocol is twice the sQNM. Also, the minimum
deconstruction cost for a variant of quantum deconstruction protocol is given
by twice the sQNM of the state.
