  We investigate prospects for the detection of high-energy neutrinos produced
in the prolonged jets of short gamma-ray bursts (sGRBs). The X-ray lightcurves
of sGRBs show extended emission components lasting for 100-1000 seconds, which
are considered to be produced by prolonged engine activity. Jets by prolonged
engine activity should interact with photons in the cocoon formed by the jet
propagation inside the ejecta of neutron star mergers. We calculate neutrino
emission from jets by prolonged engine activity, taking account of the
interaction between photons provided from the cocoon and cosmic rays
accelerated in the jets. We find that IceCube-Gen2, a future neutrino
telescope, with the second-generation gravitational wave detectors will
probably be able to observe neutrino signals associated with gravitational
waves with around 10 years of operation, regardless of the assumed value of the
Lorentz factor of the jets. Neutrino observations may enable us to constrain
the dissipation region of the jets. We apply this model to GRB 211211A, a
peculiar long GRB whose origin may be a binary neutron-star merger. Our model
predicts that IceCube is unlikely to detect any associated neutrino, but a few
similar events will be able to put a meaningful constraint on the physical
quantities of the prolonged engine activities.
