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Being a very promising technology, with impressive advances in the recent years, it is still unclear how quantum computing will scale to satisfy the requirements of its most powerful applications. Although continued progress in the fabrication and control of qubits is required, quantum computing scalability will depend as well on a comprehensive architectural design considering a multi-core approach as an alternative to the traditional monolithic version, hence including a communications perspective. However, this goes beyond introducing mere interconnects. Rather, it implies consolidating the full communications stack in the quantum computer architecture. In this paper, we propose a double full-stack architecture encompassing quantum computation and quantum communications, which we use to address the monolithic versus multi-core question with a structured design methodology. For that, we revisit the different quantum computing layers to capture and model their essence by highlighting the open design variables and performance metrics. Using behavioral models and actual measurements from existing quantum computers, the results of simulations suggest that multi-core architectures may effectively unleash the full quantum computer potential.