Is there a quantitative relation between the correlations at spacelike intervals possible in quantum field theory vs classical field theory? In quantum field theory, causality is imposed by demanding that field operators at spacetime points separated by spacelike intervals commute. However, time-ordered field correlation functions between such points can be finite, and is commonly understood to arise due to entanglement. (I read this in a comment by Gerard 't Hooft on https://www.quantamagazine.org/20160517-pilot-wave-theory-gains-experimental-support/)
However, it is possible to imagine situations with classical fields producing finite correlations at spacelike points. Imagine a spherical wave from a classical electromagnetic monochromatic point source. The electric fields at spacelike separated points on a large spherical wavefront can be perfectly correlated. 
The question then is, is there a quantitative relation between the strength of correlations at spacelike intervals possible in quantum field theory vs classical field theory? I believe this would be tantamount to a reformulation of Bell's theorem in the language of quantum field theory.
 A: The classical equivalent of spacelike commuting operators is fields with spacelike vaishing Poisson brackets - not spacelike vanishing correlators.
Note that the Wightman 2-point functions, which are the quantum analogues of classical correlation functions, also do not vanish at spacelike arguments!
A: The type of correlation that is required to violate the Bell inequality is not so much a matter of the strength of the correlation. It requires a type of correlation that is qualitatively different from the correlations in classical theories.
Here one needs to be more specific by what one means by the term correlation. Normally the term correlation refers to a two-point function $\langle \phi_1^* \phi_2 \rangle$. In quantum field theory such two-point functions give rise to propagators or mass. 
In classical theories such as stochastic optics two-point functions give the mutual coherence functions that describe the coherence properties in an optical field. In the latter case one can have correlations at two space-like separated points. This would indicate spatial coherence in the optical field. However, this does not violate any causality principles, because the correlation does not represent a causal link. Neither does it represent entanglement.
To investigate the entanglement in a state (which is a requirement to violate the Bell inequality) one needs a different kind of correlation. This is represented by a four-point function $\langle\phi_m^* \phi_n^* \phi_p \phi_q\rangle$, where $m,n,p,q$ label the different degrees of freedom in the field. This four-point function is often represented as a density matrix/operator and denoted by $\rho$. One can now used this density matrix to compute quantities such as the concurrence to quantify the amount of entanglement in the state.
There is a classical analog to quantum entanglement which, is called classical or local entanglement to distinguish it from quantum or nonlocal entanglement. In the classical case the correlation function needs to use two different degrees of freedom (e.g. polarization and spatial mode) from the same field to replace the two different fields used in the quantum case. As a consequence the classical entanglement is always local. Apart form this distinction the two concepts are formally exactly the same. One can use classical entanglement to violate a local version of the Bell inequality, but not the normal nonlocal Bell inequality.
