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With the rapid advancement of artificial intelligence (AI) and intelligent science, intelligent edge computing has been widely adopted. However, the limitations of traditional methods, such as poor adaptability and the slow convergence of heuristic algorithms, are becoming increasingly evident. To enable sustainable and resource-efficient edge applications, this paper proposes an online task offloading framework for wireless powered mobile edge computing (MEC) networks, called Quantum Attention-based Reinforcement learning for Online Offloading (QAROO). The system employs a binary offloading strategy with the aim of co-optimizing computing and energy resources in dynamic channel environments. In response to the issues of poor adaptability in traditional approaches and the slow convergence of heuristic algorithms, the framework integrates quantum neural networks and attention mechanisms, introducing three key improvements: using recurrent neural networks to enhance temporal modeling capability, proposing an uncertainty-guided quantization method to improve exploration efficiency, and incorporating attention mechanisms into quantum networks to strengthen feature representation. Experiments demonstrate that the proposed method outperforms comparative schemes in terms of normalized computation speed and processing time, offering an efficient and stable solution for online task offloading in large-scale Internet of Things (IoT) dynamic environments.