Collaboration is a key challenge in distributed multi-agent reinforcement learning (MARL) environments. Learning frameworks for these decentralized systems must weigh the benefits of explicit player coordination against the communication overhead and computational cost of sharing local observations and environmental data. Quantum computing has sparked a potential synergy between quantum entanglement and cooperation in multi-agent environments, which could enable more efficient distributed collaboration with minimal information sharing. This relationship is largely unexplored, however, as current state-of-the-art quantum MARL (QMARL) implementations rely on classical information sharing rather than entanglement over a quantum channel as a coordination medium. In contrast, in this paper, a novel framework dubbed entangled QMARL (eQMARL) is proposed. The proposed eQMARL is a distributed actor-critic framework that facilitates cooperation over a quantum channel and eliminates local observation sharing via a quantum entangled split critic. Introducing a quantum critic uniquely spread across the agents allows coupling of local observation encoders through entangled input qubits over a quantum channel, which requires no explicit sharing of local observations and reduces classical communication overhead. Further, agent policies are tuned through joint observation-value function estimation via joint quantum measurements, thereby reducing the centralized computational burden. Experimental results show that eQMARL with $\Psi^{+}$ entanglement converges to a cooperative strategy up to $17.8\\%$ faster and with a higher overall score compared to split classical and fully centralized classical and quantum baselines. The results also show that eQMARL achieves this performance with a constant factor of $25$-times fewer centralized parameters compared to the split classical baseline.

Split learning (SL) is an emergent distributed learning framework which can mitigate the computation and wireless communication overhead of federated learning. It splits a machine learning model into a device-side model and a server-side model at a cut layer. Devices only train their allocated model and transmit the activations of the cut layer to the server. However, SL can lead to data leakage as the server can reconstruct the input data using the correlation between the input and intermediate activations. Although allocating more layers to a device-side model can reduce the possibility of data leakage, this will lead to more energy consumption for resource-constrained devices and more training time for the server. Moreover, non-iid datasets across devices will reduce the convergence rate leading to increased training time. In this paper, a new personalized SL framework is proposed. For this framework, a novel approach for choosing the cut layer that can optimize the tradeoff between the energy consumption for computation and wireless transmission, training time, and data privacy is developed. In the considered framework, each device personalizes its device-side model to mitigate non-iid datasets while sharing the same server-side model for generalization. To balance the energy consumption for computation and wireless transmission, training time, and data privacy, a multiplayer bargaining problem is formulated to find the optimal cut layer between devices and the server. To solve the problem, the Kalai-Smorodinsky bargaining solution (KSBS) is obtained using the bisection method with the feasibility test. Simulation results show that the proposed personalized SL framework with the cut layer from the KSBS can achieve the optimal sum utilities by balancing the energy consumption, training time, and data privacy, and it is also robust to non-iid datasets.