  The combination of various physically plausible properties, such as no
signaling, determinism, and experimental free will, is known to be incompatible
with quantum correlations. Hence, these properties must be individually or
jointly relaxed in any model of such correlations. The necessary degrees of
relaxation are quantified here, via natural distance and information-theoretic
measures. This allows quantitative comparisons between different models in
terms of the resources, such as the number of bits, of randomness,
communication, and/or correlation, that they require. For example, measurement
dependence is a relatively strong resource for modeling singlet state
correlations, with only 1/15 of one bit of correlation required between
measurement settings and the underlying variable. It is shown how various
'relaxed' Bell inequalities may be obtained, which precisely specify the
complementary degrees of relaxation required to model any given violation of a
standard Bell inequality. The robustness of a class of Kochen-Specker theorems,
to relaxation of measurement independence, is also investigated. It is shown
that a theorem of Mermin remains valid unless measurement independence is
relaxed by 1/3. The Conway-Kochen 'free will' theorem and a result of Hardy are
less robust, failing if measurement independence is relaxed by only 6.5% and
4.5%, respectively. An appendix shows the existence of an outcome independent
model is equivalent to the existence of a deterministic model.
