  Bilayer quantum Hall (BLQH) systems, which underlie a $U(4)$ symmetry,
display unique quantum coherence effects. We study coherent states (CS) on the
complex Grassmannian $\mathbb G_2^4=U(4)/U(2)^2$, orthonormal basis, $U(4)$
generators and their matrix elements in the reproducing kernel Hilbert space
$\mathcal H_\lambda(\mathbb G_2^4)$ of analytic square-integrable holomorphic
functions on $\mathbb G_2^4$, which carries a unitary irreducible
representation of $U(4)$ with index $\lambda\in\mathbb N$. A many-body
representation of the previous construction is introduced through an oscillator
realization of the $U(4)$ Lie algebra generators in terms of eight boson
operators. This particle picture allows us for a physical interpretation of our
abstract mathematical construction in the BLQH jargon. In particular, the index
$\lambda$ is related to the number of flux quanta bound to a bi-fermion in the
composite fermion picture of Jain for fractions of the filling factor $\nu=2$.
The simpler, and better known, case of spin-$s$ CS on the Riemann-Bloch sphere
$\mathbb{S}^2=U(2)/U(1)^2$ is also treated in parallel, of which Grassmannian
$\mathbb G_2^4$-CS can be regarded as a generalized (matrix) version.
