  We suggest a way to quantify a type of macroscopic entanglement via
distillation of Greenberger-Horne-Zeilinger states by local operations and
classical communication. We analyze how this relates to an existing measure of
quantum macroscopicity based on the quantum Fisher information in several
examples. Both cluster states and Kitaev surface code states are found to not
be macroscopically quantum but can be distilled into macroscopic
superpositions. We look at these distillation protocols in more detail and ask
whether they are robust to perturbations. One key result is that
one-dimensional cluster states are not distilled robustly but
higher-dimensional cluster states are.
