Is there an analogue of the Kochen-Specker property in quantum field theory? Motivation
Entanglement and entanglement measures are traditionally defined in finite-dimensional systems. Nowadays there are very well-known definitions of entanglement measures in quantum field theories.
In finite-dimensional systems, one also has a notion of contextuality in the sense of the Kochen-Specker theorem. There are various recent work on "measures" of contextuality.
Note that one can give examples of systems that have the KS property but no entanglement.
Question
Is there an analogue of the Kochen-Specker property in quantum field theory?
Subquestion & first step:
Does it make sense to define and measure contextuality on an arbitrary square lattice?
 A: Contextuality is only partially related with entanglement. It concerns some  consequences of the fact that  observables exist which cannot be defined simultaneously. More precisely it is a property of every realistic  hidden variable theory which explains the same outcomes of measurements of a quantum theory. The Kochen Specker theorem is consequence of the Gleason theorem. The latter, to work,  needs nothing but a separable Hilbert space with dimension greater than two. Hence QFT is included straightforwardly.
A: At the semantic level (not semantic structure of theory) of language, the concepts of contextuality should be applied and emerges on the field equation.
But is it?
I don't think so.
Before contextuality, let me remind where the contextuality came from.
After quantum entanglement, there is the quantum nonlocality which works as a strange form of the computational resource.
However, in physics, this is just combination of incompatibleness of hidden variable theory, relativity and non-determinism.
Contextuality is just a spatial concentration of nonlocality.
So I believe the nonlocality of quantum field theory must be studied first.
In a sense, nonlocality does work in qft, considering interaction range on correlated system.
Here, you might want to argue something contextual, but I believe it is hard to find such on lattice like interacting particle systems.
