A central problem in quantum information theory is understanding how quantum resources can be used to communicate information more efficiently than classical resources. We introduce quantum dense network coding -- a protocol that transmits the output of a non-Boolean function to a receiver using provably half as many qubits as bits for each sender by not transmitting the entirety of the function inputs. We show this advantage requires both shared entanglement and quantum communication, is robust to noise, and the gap in success probability between quantum and classical communication can be amplified exponentially in the number of senders. Finally, we show that dense network coding gives rise to a novel, information-theoretically secure, quantum cryptographic protocol, which we call measurement-device-independent quantum key growing.

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