  Topology and geometry of a sphere create constraints for particles that lie
on its surface which they otherwise do not experience in Euclidean space.
Notably, the number of particles and the size of the system can be varied
separately, requiring a careful treatment of systems with one or several
characteristic length scales. All this can make it difficult to precisely
determine whether a particular system is in a disordered, fluid-like, or
crystal-like state. Here, we show how order transitions in systems of particles
interacting on the surface of a sphere can be detected by changes in two
hyperuniformity parameters, derived from spherical structure factor and cap
number variance. We demonstrate their use on two different systems -- solutions
of the thermal Thomson problem and particles interacting via an ultra-soft
potential of the generalized exponential model of order 4 -- each with a
distinct parameter regulating their degree of ordering. The hyperuniformity
parameters are not only able to detect the order transitions in both systems,
but also point out the clear differences in the ordered distributions in each
due to the nature of the interaction leading to them. Our study shows that
hyperuniformity analysis of particle distributions on the sphere provides a
powerful insight into fluid- and crystal-like order on the sphere.
