One-dimensional quantum spin systems continue to provide a fertile playground for uncovering exotic quantum phenomena. This thesis investigates the interplay of frustration, and quantum entanglement in spin-$S$ Heisenberg chains. By leveraging advanced tensor network algorithms, notably the Density Matrix Renormalization Group (DMRG), we compute the dynamical structure factor (DSF), which is a direct fingerprint of magnetic excitations, across various interaction regimes.\

We begin by exploring fractionalization in quantum spin chains, and revealing how bound states of magnons and their continua evolve into deconfined spinons at the first-order transition points. The thesis then develops efficient techniques to simulate time evolution in matrix product state (MPS) formalisms, enabling high-resolution spectral analysis. Applying these tools to frustrated spin-1 and spin-$3/2$ chains, we reveal spinon confinement, domain-wall dynamics, and a rich quasiparticle behavior in different quantum phases including Haldane, dimerized, and incommensurate phases.

We also extend our analysis to mixed-spin ladder systems with ferromagnetic interchain coupling, observing a crossover to effective higher-spin behavior in the excitation spectrum. In tandem with theoretical modeling, we interpret experimental results from inelastic neutron scattering (INS) from a real material \ch{$K_2Ni_2(SO_4)_3$}, and demonstrate excellent agreement with numerical predictions.\

This work bridges model Hamiltonians, numerical precision, and experimental relevance, offering novel insights into low-dimensional quantum magnetism. The methods and conclusions laid out pave the way for further investigations into emergent excitations in quantum materials.