The thesis presents a comprehensive Raman spectroscopic and first-principle density functional theory based investigation of V_(1-x)PS_(3) ; PbTaSe_(2) and 1T-TaS_(2). In V_(1-x)PS_(3),the detection of fractionalized excitations (Majorana fermions) suggests a potential Kitaev spin liquid phase, which becomes more pronounced as dimensionality decreases. Evidence of temperature-driven structural phase transition and topological surface phonons has been detected in PbTaSe_(2). TaS_(2) in 1T-phase reveals the presence of a hidden quantum state, a Mott-insulating phase, and a quantum spin liquid state. The observed phenomenon in these quantum systems makes them a promising candidate for advanced technologies such as quantum computing and communication.