  The effect of quantum steering arises from the judicious combination of an
entangled state with a set of incompatible measurements. Recently, it was shown
that this form of quantum correlations can be quantified in terms of a
dimension, leading to the notion of genuine high-dimensional steering. While
this naturally connects to the dimensionality of entanglement (Schmidt number),
we show that this effect also directly connects to a notion of dimension for
measurement incompatibility. More generally, we present a general connection
between the concepts of steering and measurement incompatibility, when
quantified in terms of dimension. From this connection, we propose a novel
twist on the problem of simulating quantum correlations. Specifically, we show
how the correlations of certain high-dimensional entangled states can be
exactly recovered using only shared randomness and lower-dimensional
entanglement. Finally, we derive criteria for testing the dimension of
measurement incompatibility, and discuss the extension of these ideas to
quantum channels.
