  We study the distinguishability norms associated to families of locally
restricted POVMs on multipartite systems. These norms (introduced by Matthews,
Wehner and Winter) quantify how quantum measurements, subject to locality
constraints, perform in the task of discriminating two multipartite quantum
states. We mainly address the following question regarding the behaviour of
these distinguishability norms in the high-dimensional regime: On a bipartite
space, what are the relative strengths of standard classes of locally
restricted measurements? We show that the class of PPT measurements typically
performs almost as well as the class of all measurements whereas restricting to
local measurements and classical communication, or even just to separable
measurements, implies a substantial loss. We also provide examples of state
pairs which can be perfectly distinguished by local measurements if (one-way)
classical communication is allowed between the parties, but very poorly without
it. Finally, we study how many POVMs are needed to distinguish almost perfectly
any pair of states on $\mathbf{C}^d$, showing that the answer is
$\exp(\Theta(d^2))$.
