  Multi-photon entangled states of light are key to advancing quantum
communication, computation, and metrology. Current methods for building such
states are based on stitching together photons from probabilistic sources. The
probability of $N$ such sources firing simultaneously decreases exponentially
with $N$, imposing severe limitations on the practically achievable number of
coincident photons. We tackle this challenge with a quantum interference buffer
(QIB), which combines three functionalities: firstly, it stores polarization
qubits, enabling the use of polarization-entangled states as resource;
secondly, it implements entangled-source multiplexing, greatly enhancing the
resource-state generation rates; thirdly, it implements time-multiplexed,
on-demand linear optical networks for interfering subsequent states. Using the
QIB, we multiplex 21 Bell-state sources and demonstrate a nine-fold enhancement
in the generation rate of four-photon GHZ states. The enhancement scales
exponentially with the photon number; larger states benefit more strongly.
Multiplexed photon entanglement and interference will find diverse applications
in quantum photonics, allowing for practical realisations of multi-photon
protocols.
