Topological superconduc
tors provide a promising route to fault-tolerant quantum computing\; however\, it prov
ed hard to find or engineer them. Recently\, topological superconductivity
was predicted to arise at the interface between quantum Hall and conventi
onal superconducting states. Since both ingredients are readily available
in the lab\, topological superconductivity seemed to be within the reach.
The predictions\, however\, focus on the idealized &ldquo\;clean&rdquo\; c
ase\, whereas only *strongly disordered* superconductors are compat
ible with high magnetic fields needed for the quantum Hall effect.<
/span> Can topological sup
erconductivity survive the presence of disorder?

\;

\n\nWe develop a theory of two counter-propagating quantum H
all edge states coupled via a narrow* disordered *superconductor. W
e show that\, in contrast to the clean-case predictions\, the edge states
do not turn into a topological superconductor. Instead\, the disorder tune
s them to the critical point between the trivial insulating phase and the
topological phase. We determine the manifestations of this criticality in
the charge transport\, finding that the critical conductance is a random\,
sample-specific quantity with a zero average and unusual bias dependence.
The developed theory of disordered superconductor-quantum Hall interfaces
offers an interpretation of recent experiments.