  We develop a protocol for entanglement generation in the quantum internet
that allows a repeater node to use $n$-qubit Greenberger-Horne-Zeilinger (GHZ)
projective measurements that can fuse $n$ successfully-entangled {\em links},
i.e., two-qubit entangled Bell pairs shared across $n$ network edges, incident
at that node. Implementing $n$-fusion, for $n \ge 3$, is in principle not much
harder than $2$-fusions (Bell-basis measurements) in solid-state qubit
memories. If we allow even $3$-fusions at the nodes, we find---by developing a
connection to a modified version of the site-bond percolation problem---that
despite lossy (hence probabilistic) link-level entanglement generation, and
probabilistic success of the fusion measurements at nodes, one can generate
entanglement between end parties Alice and Bob at a rate that stays constant as
the distance between them increases. We prove that this powerful network
property is not possible to attain with any quantum networking protocol built
with Bell measurements and multiplexing alone. We also design a two-party
quantum key distribution protocol that converts the entangled states shared
between two nodes into a shared secret, at a key generation rate that is
independent of the distance between the two parties.
