  It has been shown that the conditional probability distributions obtained by
performing measurements on an uncharacterized physical system can be used to
infer its underlying dimension in a device-independent way both in the
classical and quantum setting. We analyze several aspects of the structure of
the sets of probability distributions corresponding to a certain dimension
taking into account whether shared randomness is available as a resource or
not. We first consider the so-called prepare-and-measure scenario. We show that
quantumness and shared randomness are not comparable resources. That is, on the
one hand, there exist behaviours that require a quantum system of arbitrarily
large dimension in order to be observed while they can be reproduced with a
classical physical system of minimal dimension together with shared randomness.
On the other hand, there exist behaviours which require exponentially larger
dimensions classically than quantumly even if the former is supplemented with
shared randomness. We also show that in the absence of shared randomness, the
sets corresponding to a sufficiently small dimension are negligible
(zero-measure and nowhere dense) both classically and quantumly. This is in
sharp contrast to the situation in which this resource is available and
explains the exceptional robustness of dimension witnesses in the setting in
which devices can be taken to be uncorrelated. We finally consider the Bell
scenario in the absence of shared randomness and prove some non-convexity and
negligibility properties of these sets for sufficiently small dimensions. This
shows again the enormous difference induced by the availability or not of this
resource.
