  In this work we introduce an intermediate setting between quantum nonlocality
and communication complexity problems. More precisely, we study the value of
XOR games $G$ when Alice and Bob are allowed to use a limited amount of one-way
classical communication $\omega_{o.w.-c}(G)$ (resp. one-way quantum
communication $\omega_{o.w.-c}^*(G)$), where $c$ denotes the number of bits
(resp. qubits). The key quantity here is the quotient
$\omega_{o.w.-c}^*(G)/\omega_{o.w.-c}(G)$.
  We provide a universal way to obtain Bell inequality violations of general
Bell functionals from XOR games for which the quotient
$\omega_{o.w.-c}^*(G)/\omega_{o.w.-2c}(G)$ is larger than 1. This allows, in
particular, to find (unbounded) Bell inequality violations from communication
complexity problems in the same spirit as the recent work by Buhrman et al.
(2016).
  We also provide an example of a XOR game for which the previous quotient is
optimal (up to a logarithmic factor) in terms of the amount of information $c$.
Interestingly, this game has only polynomially many inputs per player. For the
related problem of separating the classical vs quantum communication complexity
of a function, the known examples attaining exponential separation require
exponentially many inputs per party.
