While trying to rework an upcoming quiz problem for a quantum physics course, I came up with the following question which turned out to be harder than I expected. (Note: I take $\hbar =1$ in all statements below.)
I prepare a spin-$S$ state with maximum $z$-component i.e. $s_z=s$, and measure the spin components in the plane; first $s_x$, and then $s_y$. I can then repeat this pair of measurements ad infinitum, obtaining a sequence of points in the $(s_x,s_y)$ plane.
For small $S$, one can compare the possible outcomes by hand; indeed, for the simplest case $s=1/2$, there are four outcomes $(s_x,s_y)=(\pm\frac12,\pm\frac12)$ and by symmetry I expect each of them to occur with probability 1/4. This symmetry is lost for $S>1/2$, but some relatively explicit properties should be available at small $S$.
If $S$ is large, on the other hand, I would expect $(s_x,s_y)$ is more likely to be found at some finite distance from the origin. This intuition is motivated by the fact that the spin operators satisfy $$S_x^2+S_y^2 = S^2-S_z^2=s(s+1)-s^2=s,$$ suggesting some preferred radius in spin space. So the question I would pose is:
What is the probability distribution function of this sequence? In particular, how does it behave for large $S$?