Due to the intensifying impacts of extreme climate changes, drought forecasting (DF), which aims to predict droughts from historical meteorological data, has become increasingly critical for monitoring and managing water resources. Though drought conditions often exhibit spatial climatic coherence among neighboring regions, benchmark deep learning-based DF methods overlook this fact and predict the conditions on a region-by-region basis. Using the Standardized Precipitation Evapotranspiration Index (SPEI), we designed and trained a novel and transformative spatially-aware DF neural network, which effectively captures local interactions among neighboring regions, resulting in enhanced spatial coherence and prediction accuracy. As DF also requires sophisticated temporal analysis, the Mamba network, recognized as the most accurate and efficient existing time-sequence modeling, was adopted to extract temporal features from short-term time frames. We also adopted quantum neural networks (QNN) to entangle the spatial features of different time instances, leading to refined spatiotemporal features of seven different meteorological variables for effectively identifying short-term climate fluctuations. In the last stage of our proposed SPEI-driven quantum spatially-aware Mamba network (SQUARE-Mamba), the extracted spatiotemporal features of seven different meteorological variables were fused to achieve more accurate DF. Validation experiments across El Niño, La Niña, and normal years demonstrated the superiority of the proposed SQUARE-Mamba, remarkably achieving an average improvement of more than 9.8% in the coefficient of determination index (R^2) compared to baseline methods, thereby illustrating the promising roles of the temporal quantum entanglement and Mamba temporal analysis to achieve more accurate DF.