Clock matrix vs matrix clock

Question

In the process of research leading up to my previous question, I found out about matrix, vector & logical clocks.

The citation in the aforementioned question mentions clock and shift matrices. Wikipedia states:

These two matrices are also the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces as formulated by Hermann Weyl, and find routine applications in numerous areas of mathematical physics. The clock matrix amounts to the exponential of position in a "clock" of d hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg on a d-dimensional Hilbert space.

I am curious to find out

if there is any usage of matrix, vector or logical clocks in quantum computing
how clock matrices and matrix clocks compare & contrast

Related: Are the 'clock' and 'shift' operators used in Qudit codes physically realizable? — Sanchayan Dutta, Dec 18 '18 at 02:49
do you have any reason to believe that there is any relationship between "clock matrices" and "matrix clocks"? Because it looks to me like there is none whatsoever. "Clock matrix" is just a name, they've got nothing to do with time. Clock matrices are useful in that they are a direct generalisation of the Pauli $Z$ matrix to higher dimensions. Matrix clocks on the other hand, is a term I've never seen outside of that Wikipedia page — glS, Jun 05 '19 at 09:08

Clock matrix vs matrix clock

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