  We study the quantumness of correlations for ensembles of bi- and
multi-partite systems and relate it to the task of quantum data hiding.
Quantumness is here intended in the sense of minimum average disturbance under
local measurements. We consider a very general framework, but focus on local
complete von Neumann measurements as cause of the disturbance, and, later on,
on the trace-distance as quantifier of the disturbance. We discuss connections
with entanglement and previously defined notions of quantumness of
correlations. We prove that a large class of quantifiers of the quantumness of
correlations are entanglement monotones for pure bipartite states. In
particular, we define an entanglement of disturbance for pure states, for which
we give an analytical expression. Such a measure coincides with negativity and
concurrence for the case of two qubits. We compute general bounds on
disturbance for both single states and ensembles, and consider several
examples, including the uniform Haar ensemble of pure states, and pairs of
qubit states. Finally, we show that the notion of ensemble quantumness of
correlations is most relevant in quantum data hiding. Indeed, while it is known
that entanglement is not necessary for a good quantum data hiding scheme, we
prove that ensemble quantumness of correlations is.
