Not in this field directly, but first of all, talking about the "geometry of space" is more than just saying there are three spatial dimensions: geometry involv

Not in this field directly, but first of all, talking about the "geometry of space" is more than just saying there are three spatial dimensions: geometry involves the local curvature of the object. Historically the Pauli matrices are discovered by assuming certain symmetries of spacetime. This paper shows the other direction also makes sense: if we assume certain structure on quantum observables, measurable only by temporal measurements and independent of the content of the quantum state (i.e., a measurement of any system will do), we can get the spatial symmetries we want. I suppose the ideal outcome is that there is some sort of exotic algebra of observables which is well motivated somehow by purely quantum considerations and by serendipity induces all the usual spacetime symmetries + extra stuff we didn't know about. This paper itself is cute but not sure if it's very impactful, I would defer to domain experts.