Does spin 3/2 imply 2/3 full rotations? In this Wikipedia page it writes:

... a spin of 1/2 means that the particle must be rotated by two full
turns (through 720°) before it has the same configuration as when it
started.

The animation in the page demonstrates this statement (I think it is an analogy tough).
So the question is does a spin of 3/2 imply a turn of 240 degrees with the same approach?
In another Wikipedia page the mathematical expression of higher spins are given, but they look quite complicated. It would be very nice if you could explain these expressions and their physical meanings as simple as possible. Thank you.
 A: No...  Not if you are asking about the period of the rotation matrix in the quartet representation.
You know that for the quartet representation, $j=3/2$, rotations, the group element is a 4×4 rotation matrix,
$$
e^{i \theta (\hat{\boldsymbol{n}}\cdot\boldsymbol{J})} = I_4 \cos (\theta/2)\left(1+\tfrac{1}{2}\sin^2
(\theta/2)\right)+(2i \hat{\boldsymbol{n}}\cdot\boldsymbol{J} ~\sin (\theta/2))\left(1+\tfrac{1}{6} \sin^2 (\theta/2)
\right)
\\
\phantom{e^{i \theta (\hat{\boldsymbol{n}}\cdot\boldsymbol{J})}=}{}
+\frac{1}{2!} \bigl (2i \hat{\boldsymbol{n}}\cdot\boldsymbol{J}~\sin (\theta/2) \bigr)^2 \cos (\theta/2)+\frac {1}{3!}
\bigl (2i \hat{\boldsymbol{n}}\cdot\boldsymbol{J}~\sin (\theta/2) \bigr)^3.
 $$
whose period is 4π,  like all half-integral spin representations. You generalized wrong.
All even-dimensional ($2j+1$ even) representations of SU(2) have period 4π, easily read off the exponential of $J_z$.
A: Yes. The spin part of the wave function is $$e^{im_z\phi} \,.$$ It is invariant under $2\pi/m_z$ rotations about the quantization axis. The rotation angle depends on $m_z $ rather than on $S$. For a quartet state with $m_z=\pm 3/2$ this implies invariance under a (multiple of) $4\pi/3=240^\text{o}$ rotation(s). This includes $4\pi$ rotations. Quartet or any other states with $m_z=\pm 1/2$ are invariant under $4\pi$ rotations.
Invariance of all states of the quartet are requires an angle of $4\pi$. However the OP did not state that his/her question is limited to this. Therefore the present answer is the more general one.
