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The Quantum Internet, by enabling quantum communications among remote quantum nodes, is a network capable of supporting functionalities with no direct counterpart in the classical world. Indeed, with the network and communications functionalities provided by the Quantum Internet, remote quantum devices can communicate and cooperate for solving challenging computational tasks by adopting a distributed computing approach. The aim of this study is to provide the reader with an overview about the main challenges and open problems arising in the design of a distributed quantum computing ecosystem. For this, the authors provide a survey, following a bottom-up approach, from a communications engineering perspective. They start by introducing the Quantum Internet as the fundamental underlying infrastructure of the distributed quantum computing ecosystem. Then they go further, by elaborating on a high-level system abstraction of the distributed quantum computing ecosystem. They describe such an abstraction through a set of logical layers. Thereby, they clarify dependencies among the aforementioned layers and, at the same time, a road-map emerges.