We explore different implementations of the quantum Bell causality (QBC) condition in the quantum sequential growth (QSG) dynamics of causal set quantum gravity, for noncommuting transition operators. Assuming a nonsingular dynamics we show that for the two most natural choices of operator orderings for the QBC, the transition operator algebra $\mathcal{A}$ reduces to a commutative one. As a third choice, we take the operator ordering to depend on the size of the precursor set. We find several new commutation relations which further constrain $\mathcal{A}$ but do not imply commutativity. On the other hand, if any of the generators of the “antichain subalgebra” $\mathcal{Q}$ belongs to its center, then this implies commutativity of $\mathcal{A}$. The complexity of the algebra prevents us from obtaining a general form for the transition operators, which hinders computability. In an attempt to construct the simplest nontrivial $d=2$ representation of $\mathcal{A}$, we find that a Pauli matrix representation of the generators of $\mathcal{Q}$ leads to inconsistencies. Our work can be viewed as a first step toward finding a noncommutative realization of QSG.

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• Quantum cosmology
• Quantum gravity