The advent of artificial intelligence (AI) and edge computing technologies has ushered in a new era for automatic modulation classification (AMC), with a particular focus on deep learning (DL)-based approaches. These AI-based AMC systems, deployed at edge devices, have shown remarkable advancements in recognizing and classifying a wide range of wireless signals. However, a significant challenge in this domain lies in the lack of adequate explanations for the models employed, which, in turn, hampers model accuracy and training efficiency. Consequently, the practical applications and further enhancements of these models are constrained. To address this limitation, researchers have started to explore interpretable methods for the technical analysis of DL-based AMC systems. In this paper, we delve into the subject, leveraging recent developments in interpretable methods, to provide a comprehensive review of applicable techniques and to outline the existing research challenges in the field of interpretable AMC. Moreover, we introduce a novel interpretable AI-based AMC framework designed to augment the interpretability of AMC outcomes. This framework achieves this by elucidating the contribution of wireless signal features to the training process of DL networks. In addition, we explore the potential impact of quantum computing on AMC systems. Quantum computing, with its ability to process complex computations at unprecedented speeds, offers a promising avenue for enhancing the efficiency and accuracy of AMC. By integrating quantum algorithms with AI-based AMC frameworks, it is possible to significantly improve the performance of wireless signal classification, especially in environments with high signal variability and noise. Our experimental results demonstrate that the proposed approach possesses the capacity to unravel the classification mechanisms hidden within opaque auto-modulated classification networks. Furthermore, it exhibits the potential to facilitate the training of networks on edge devices with reduced energy consumption and enhanced classification accuracy, thereby addressing critical challenges in the field of wireless signal classification. Additionally, the integration of quantum computing techniques holds the promise of further advancing the capabilities of AMC systems, paving the way for more robust and efficient wireless communication technologies.