  Theories involving localized collapse allow the possibility that classical
information could be obtained about quantum states without using POVMS and
without allowing superluminal signalling. We can model this by extending
quantum theory to include hypothetical devices that read out information about
the local quantum state at a given point, defined by considering only collapses
in its past light cone. Like Popescu-Rohrlich boxes, these hypothetical devices
would have practical and scientific implications if realisable. These include
signalling through opaque media, probing the physics of distant or opaque
systems without needing a reflected signal and giving detailed information
about collapse dynamics without requiring direct observation of the collapsing
system. These potential applications motivate systematic searches for possible
signatures of these nonstandard extensions of quantum theory, and in particular
for relevant gravitational effects, such as the validity of semi-classical
gravity on small scales.
