  Classical and quantum physics provide fundamentally different predictions
about experiments with separate observers that do not communicate, a phenomenon
known as quantum nonlocality. This insight is a key element of our present
understanding of quantum physics, and also enables a number of information
processing protocols with security beyond what is classically attainable.
Relaxing the pivotal assumption of no communication leads to new insights into
the nature quantum correlations, and may enable new applications where security
can be established under less strict assumptions. Here, we study such
relaxations where different forms of communication are allowed. We consider
communication of inputs, outputs, and of a message between the parties. Using
several measures, we study how much communication is required for classical
models to reproduce quantum or general no-signalling correlations, as well as
how quantum models can be augmented with classical communication to reproduce
no-signalling correlations.
