  In a quantum network that successfully creates links, shared Bell states
between neighboring repeater nodes, with probability $p$ in each time slot, and
performs Bell State Measurements at nodes with success probability $q<1$, the
end to end entanglement generation rate drops exponentially with the distance
between consumers, despite multi-path routing. If repeaters can perform
multi-qubit projective measurements in the GHZ basis that succeed with
probability $q$, the rate does not change with distance in a certain $(p,q)$
region, but decays exponentially outside. This region where the distance
independent rate occurs is the supercritical region of a new percolation
problem. We extend this GHZ protocol to incorporate a time-multiplexing
blocklength $k$, the number of time slots over which a repeater can
mix-and-match successful links to perform fusion on. As $k$ increases, the
supercritical region expands. For a given $(p,q)$, the entanglement rate
initially increases with $k$, and once inside the supercritical region for a
high enough $k$, it decays as $1/k$ GHZ states per time slot. When memory
coherence time exponentially distributed with mean $\mu$ is incorporated, it is
seen that increasing $k$ does not indefinitely increase the supercritical
region; it has a hard $\mu$ dependent limit. Finally, we find that
incorporating space-division multiplexing, i.e., running the above protocol
independently in up to $d$ disconnected network regions, where $d$ is the
network's node degree, one can go beyond the 1 GHZ state per time slot rate
that the above randomized local link-state protocol cannot surpass. As $(p,q)$
increases, one can approach the ultimate min-cut entanglement generation
capacity of $d$ GHZ states per slot.
