Looking for a refuge from election anxiety? Let me tell you about the Majorana representation and coherent spin states. I just learned about them and I think they're very beautiful.

Any n-qubit state |Φ> that lives in the symmetric subspace can be obtained by symmetrizing some product state |ψ₁>⊗⋯⊗|ψₙ> over all n! qubit permutations (and then normalizing it).

The reverse map (from the symmetric subspace to a product state) assigns to each symmetric n-qubit state |Φ> a set of n single-qubit states |ψₖ> which can be pictured as n points on the Bloch sphere. These n states are the called Majorana representation of |Φ>.

Note that a qubit state |ψ> = a|0> + b|1> is uniquely determined by a single complex number, the ratio z = a/b. Indeed, this is because |ψ> is normalized and we don't care about its global phase.

It turns out that you can compute the Majorana representation by finding the roots z₁,…,zₙ of a certain polynomial (the Majorana polynomial). They in turn determine the n single-qubit states |ψ₁>,…,|ψₙ>.

You can read more about this here:
https://arxiv.org/abs/0910.0630 

What is cool about this is that applying U on each qubit of the symmetric state |Φ> corresponds to applying U to each of the n single-qubit states |ψₖ> of its Majorana representation.

In fact, the action of U⊗⋯⊗U in the n-qubit symmetric subspace is an irreducible representation of U(2). It has dimension n+1 since the symmetric subspace is spanned by the symmetrizations of |0…00>, |0…01>, |0…11>, …, |1…11> (there are n+1 different Hamming weights).

Tensor power states |ψ>^⊗n play a special role since they are already symmetric under qubit permutations. Their Majorana representation consists of n identical states |ψ>. Also, tensor power states are the orbit of |0>^⊗n under U^⊗n.

If you shrink down the n-qubit sym. subspace to a single qudit of dim. n+1, the tensor power states become what is known as coherent spin states. They are the orbit of |0> under the (n+1)-dim irrep of U(2). They are a finite-dim. analogue of coherent states in quantum optics.

You can read more about coherent spin states in Radcliffe's paper:
https://phas.ubc.ca/~stamp/TEACHING/PHYS501/NOTES/FILES/Radcliffe_SpinCoh--JPA71.pdf
or in Arecchi et al.:
https://doi.org/10.1103/PhysRevA.6.2211
Here's an extensive Rev. Mod. Phys. survey on coherent states, including a section on "atomic coherent states":
https://doi.org/10.1103/RevModPhys.62.867