  Comprehending the connections between the geometric, topological, and
dynamical structures of integrable quantum systems with quantum phenomena
exploitable in quantum information tasks, such as quantum entanglement, is a
major problem in geometric information science. In this work we investigate
these issues in a physical system of $\mathcal{N}$ interacting
spin-$\mathtt{s}$ under long-range Ising model. We discover the relevant
dynamics, identify the corresponding quantum phase space and we derive the
associated Fubini-Study metric. Through the application of the Gauss-Bonnet
theorem and the derivation of the Gaussian curvature, we have proved that the
dynamics occurs on a spherical topology manifold. Afterwards, we analyze the
gained geometrical phase under the arbitrary and cyclic evolution processes and
solve the quantum brachistochrone problem by establishing the time-optimal
evolution. Moreover, by narrowing the system to a two spin-$\mathtt{s}$ system,
we explore the relevant entanglement from two different perspectives; The first
is geometrical in nature and involves the investigation of the interplay
between the entanglement degree and the geometrical structures, such as the
Fubini-Study metric, the Gaussian curvature and the geometrical phase. The
second is dynamical in nature and tackles the entanglement effect on the
evolution speed and geodesic distance. Additionally, we resolve the quantum
brachistochrone problem based on the entanglement degree.
