In my review of "Foundations of quantum physics II. The thermal interpretation", I uncritically claimed that the collection of all q-expectations and q-correlations would be a state in the thermal interpretation, without noticing that timelike q-correlations could be problematic:
The states of a system in the thermal interpretation are encoded by density operators. However, a state itself is rather the collection of all q-expectations for that state. Section "4.1 Beables and observability in quantum field theory" states this as follows:
"According to the thermal interpretation, there is nothing in quantum field theory apart from q-expectations of the fields and q-correlations. The quantities accessible to an observer are those q-expectations and q-correlations whose arguments are restricted to the observer’s world tube. More precisely, what we can observe is contained in the least oscillating contributions to these q-expectations and q-correlations. The spatial and temporal high frequency part is unobservable due to the limited resolution of our instruments."
This quote is relevant for a number of reasons. The q-expectations have a spatial and temporal dependence (as parameters). The q-correlations even depend on more than one different spatial and temporal parameter. (This is how I interpret the difference between q-expectations and q-correlations in this quote.) We are given an explicit reason why some q-expectations are not observable (because their high frequency dependence of the spatial and temporal parameters exceed the resolution limits of our instruments).
My claim is problematic for (non-qft) quantum mechanics, because I explicitly claimed that the state is encoded by a density operator, and at the same time claim that q-correlation with two or more temporal parameters would be part of the state. But the density matrix at a given time could only encode those "two times q-correlations" if it additionally used some Hamiltonian in a Schrödinger equation to compute those. But then that Hamiltonian would have to be part of that encoding too.
I think that timelike q-correlations might be problematic (as state) in quantum field theory too, because it is unclear which measurement setup should be able to measure even their lowest frequency components. (But maybe those timelike q-correlations would never have low frequency components anyway, by an effect similar to evanescent modes. But even in this case, it would be nice if the situation could be further clarified.)