This is an answer from the Physics point of view. The Bloch sphere is a kind of "phase space" for the qubit. Now let's see what is a phase space. Phase space typically appears in Classical Mechanics in order to study the dynamics of a classical system. You can do lots of things there, for instance study chaotic systems.
How do you do it in Quantum Mechanics? Historically much of the development has been aimed to study for instance light (technically is the harmonic oscillator but it doesn't matter much). You can have your master equation with dissipation and temperature and study the solutions of $\partial_t \rho = \mathcal{L}[\rho]$. Another view is to map this equation onto the phase space (most of the time it is tricky!) using e.g. the Wigner Function. There you can see how your system behaves as a function of time and what does it do because of the temperature, etc.
This is all well because the harmonic oscillator can be described in principle in continous $(x,p)$ variables. On the other hand, qubits are somehow "discrete", so it is not at all trivial to define a mapping $\rho_{\rm qubits}\rightarrow W(\rho_{\rm qubits})$ over the phase space because one has to fulfill several requierements (!)
Many theoretical uses can be found in the literature, as well as recent developments on how to define a reasonable phase space function for qubits. I leave some references here:
[1] W. P. Schleich. Quantum Optics in Phase Space. 1st ed. Wiley-VCH (2001). (Classical treament of Phase Space for Quantum Mechanics.)
[2] A. Perelomov. Generalized Coherent States and Their Applications. 1986th ed. Springer-Verlag (1986). (You can find partial answers to your questions here!)
[3] C. Muñoz, I. Sainz and A. B. Klimov. Macroscopic approach to N-qudit systems. J. Phys. A: Math. Theor. 53 245302(2020). (This is a recent example of how e.g. to see and use the "phase space" for qudits!)
