The ability to model and forecast financial time series is essential for modern economic strategy, guiding critical decisions in risk management, asset allocation, and derivative pricing. While traditional econometric models, such as Autoregressive Integrated Moving Average (ARIMA) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH), perform well in stable conditions, they often fail to capture the abrupt regime shifts that characterise real-world markets. Detecting these structural breaks as they emerge, rather than a posteriori, remains a major challenge for effective forecasting. This thesis investigates how quantum mechanics can offer both a novel mathematical framework and a computational advantage to address this challenge. The financial market, with its vast number of interacting agents, inherent uncertainty, and rapid shifts in collective sentiment, provides a fertile domain for applying quantum-inspired methods.