  Let $\mathscr{C}$ be the category of finite-dimensional modules over a
simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any
height function $\xi$ and $\ell\in \mathbb{Z}_{\geq 1}$, we introduce certain
subcategories $\mathscr{C}^{\leq \xi}_\ell$ of $\mathscr{C}$, and prove that
the quantum Grothendieck ring $K_t(\mathscr{C}^{\leq \xi}_\ell)$ of
$\mathscr{C}^{\leq \xi}_\ell$ admits a quantum cluster algebra structure. Using
$F$-polynomials and monoidal categorifications of cluster algebras, we classify
all real simple modules in $\mathscr{C}^{\leq \xi}_1$ in terms of their highest
$\ell$-weight monomials, among them the families of type $D$ and type $E$ are
new. For any $\ell$, inspired by Hernandez and Leclerc's work, we propose two
conjectures for the study of real simple modules, and prove them for the
subcategories $\mathscr{C}^{\leq \xi}_\ell$ whose Grothendieck rings are
cluster algebras of finite type.
