Spin states of maximal projection along some direction in space are called (spin) coherent, and are, in many aspects, the "most classical" available. For any spin s, the spin coherent states form a 2-sphere in the projective Hilbert space P of the system. We address several questions regarding that sphere, in particular its possible intersections with complex lines. We also find that, like Dali's iconic clocks, it extends in all possible directions in P. We give a simple expression for the Majorana constellation of the linear combination of two coherent states, and use Mason's theorem to give a lower bound on the number of distinct stars of a linear combination of two arbitrary spin-s states. Finally, we plot the image of the spin coherent sphere, assuming light in P propagates along Fubini-Study geodesics. We argue that, apart from their intrinsic geometric interest, such questions translate into statements experimentalists might find useful.

We generalize the Majorana stellar representation of spin-s pure states to mixed states, and in general to any hermitian operator, defining a bijective correspondence between three spaces: the spin density-matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state ρ. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-s states described in [1].

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We present a quantum description of the mechanism by which a free-falling cat manages to reorient itself and land on its feet, having all along zero angular momentum. Our approach is geometrical, making use of the fiber bundle structure of the cat configuration space. We show how the classical picture can be recovered, but also point out a purely quantum scenario, that ends up with a Schroedinger cat. Finally, we sketch possible applications to molecular, nuclear, and nano-systems.