  The $SO(5)$ Landau model is the mathematical platform of the 4D quantum Hall
effect and provide a rare opportunity for a physical realization of the fuzzy
four-sphere. We present an integrated analysis of the $SO(5)$ Landau models and
the associated matrix geometries through the Landau level projection. With the
$SO(5)$ monopole harmonics, we explicitly derive matrix geometry of a
four-sphere in any Landau level: In the lowest Landau level the matrix
coordinates are given by the generalized $SO(5)$ gamma matrices of the fuzzy
four-sphere satisfying the quantum Nambu algebra, while in higher Landau levels
the matrix geometry becomes a nested fuzzy structure realizing a pure quantum
geometry with no counterpart in classical geometry. The internal fuzzy geometry
structure is discussed in the view of an $SO(4)$ Pauli-Schr\"odinger model and
the $SO(4)$ Landau model, where we unveil a hidden singular gauge
transformation between their background non-Abelian field configurations.
Relativistic versions of the $SO(5)$ Landau model are also investigated and
relationship to the Berezin-Toeplitz quantization is clarified. We finally
discuss the matrix geometry of the Landau models in even higher dimensions.
