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The Quantum Internet is commonly envisioned as a network that transports quantum states and distributes entanglement as the key resource underpinning distributed quantum applications. This view is necessary but incomplete if quantum phenomena are confined to what the network carries or manages, while network functionalities remain grounded predominantly in classical abstractions. This Perspective advances a broader thesis: quantum phenomena can also participate directly in the realization of network functionalities. We first establish the architectural baseline by introducing entanglement as a stateful network resource whose operational condition is non-locally distributed across remote nodes, evolves through manipulation or consumption, and degrades over time through decoherence. Its evolving state must therefore remain visible to the functionalities acting on it. We then move beyond resource visibility and use two mechanisms as proof points of a broader quantum-native functional design space: entanglement generation during propagation through coherent spatial-path superposition, and operational connectivity induced by shared entanglement that can depart from the physical graph. These examples show that paths, neighbors, or participating nodes may become operationally relevant in ways determined by quantum coherence and shared entanglement, rather than by physical connectivity alone. We identify Quantum Addressing as a transversal abstraction capable of representing such relations while complementing, rather than replacing, classical addressing and signaling. The resulting perspective broadens the Quantum Internet from transporting quantum information and distributing entanglement to exploiting quantum phenomena in network functioning, opening a coupled frontier spanning routing, forwarding, control, and functionality placement.