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This study investigates the efficacy of a hybrid quantum-classical model, denoted as HQCM-EBTC, for the automated classification of brain tumors, comparing its performance against a classical counterpart. A comprehensive dataset comprising 7,576 magnetic resonance imaging (MRI) images, encompassing normal brain structures, meningioma, glioma, and pituitary tumors, was employed. The HQCM-EBTC model integrates a quantum processing layer, characterized by 5 qubits per circuit, a circuit depth of 2, and 5 parallel circuits, trained via the AdamW optimizer with a composite loss function that combines cross-entropy and attention consistency losses. The results demonstrate that HQCM-EBTC significantly outperforms the classical model, achieving an overall classification accuracy of 96.48\% compared to 86.72\%. The quantum-enhanced model exhibits superior precision, recall, and F1-scores across all tumor classes, particularly in the classification of glioma. t-SNE visualizations reveal enhanced feature separability within the quantum processing layer, leading to more distinct decision boundaries. Confusion matrix analysis further substantiates a reduction in misclassification rates with HQCM-EBTC. Moreover, attention map analysis, quantified using the Jaccard Index, indicates that HQCM-EBTC produces more localized and accurate tumor region activations, especially at higher confidence thresholds. These findings underscore the potential of quantum-enhanced models to improve brain tumor classification accuracy and localization, offering promising advancements for clinical diagnostic applications. The demonstrated ability of HQCM-EBTC to achieve higher accuracy and more precise tumor localization suggests a significant step forward in the application of quantum computing to medical imaging analysis.