  Linear constraint systems (LCS) have proven to be a surprisingly prolific
tool in the study of non-classical correlations and various related issues in
quantum foundations. Many results are known for the Boolean case, yet the
generalisation to systems of odd dimension is largely open. In particular, it
is not known whether there exist LCS in odd dimension, which admit
finite-dimensional quantum, but no classical solutions.
  Here, we approach this question from a computational perspective. We observe
that every deterministic, non-adaptive measurement-based quantum computation
(MBQC) with linear side-processing defines a LCS. Moreover, the measurement
operators of such a MBQC almost define a quantum solution to the respective
LCS: the only difference is that measurement operators generally only commute
with respect to the resource state of the MBQC. This raises the question
whether this state-dependence can be lifted in certain cases, thus providing
examples of quantum solutions to LCS in odd dimension. Our main result asserts
that no such examples arise within a large extension of the Pauli group for p
odd prime, which naturally arises from and is universal for computation in
deterministic, non-adaptive MBQC with linear side-processing.
