  We analyze a family of measures of general quantum correlations for composite
systems, defined in terms of the bipartite entanglement necessarily created
between systems and apparatuses during local measurements. For every
entanglement monotone $E$, this operational correspondence provides a different
measure $Q_E$ of quantum correlations. Examples of such measures are the
relative entropy of quantumness, the quantum deficit, and the negativity of
quantumness. In general, we prove that any so defined quantum correlation
measure is always greater than (or equal to) the corresponding entanglement
between the subsystems, $Q_E \ge E$, for arbitrary states of composite quantum
systems. We analyze qualitatively and quantitatively the flow of correlations
in iterated measurements, showing that general quantum correlations and
entanglement can never decrease along von Neumann chains, and that genuine
multipartite entanglement in the initial state of the observed system always
gives rise to genuine multipartite entanglement among all subsystems and all
measurement apparatuses at any level in the chain. Our results provide a
comprehensive framework to understand and quantify general quantum correlations
in multipartite states.
