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Classical self-supervised networks suffer from convergence problems and reduced segmentation accuracy due to forceful termination. Qubits or bilevel quantum bits often describe quantum neural network models. In this article, a novel self-supervised shallow learning network model exploiting the sophisticated three-level qutrit-inspired quantum information system, referred to as quantum fully self-supervised neural network (QFS-Net), is presented for automated segmentation of brain magnetic resonance (MR) images. The QFS-Net model comprises a trinity of a layered structure of qutrits interconnected through parametric Hadamard gates using an eight-connected second-order neighborhood-based topology. The nonlinear transformation of the qutrit states allows the underlying quantum neural network model to encode the quantum states, thereby enabling a faster self-organized counterpropagation of these states between the layers without supervision. The suggested QFS-Net model is tailored and extensively validated on the Cancer Imaging Archive (TCIA) dataset collected from the Nature repository. The experimental results are also compared with state-of-the-art supervised (U-Net and URes-Net architectures) and the self-supervised QIS-Net model and its classical counterpart. Results shed promising segmented outcomes in detecting tumors in terms of dice similarity and accuracy with minimum human intervention and computational resources. The proposed QFS-Net is also investigated on natural gray-scale images from the Berkeley segmentation dataset and yields promising outcomes in segmentation, thereby demonstrating the robustness of the QFS-Net model. [ABSTRACT FROM AUTHOR]