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We investigate prolonged engine activities of short gamma-ray bursts (SGRBs),such as extended and/or plateau emissions, as high-energy gamma-raycounterparts to gravitational waves (GWs). Binary neutron-star mergers lead torelativistic jets and merger ejecta with $r$-process nucleosynthesis, which areobserved as SGRBs and kilonovae/macronovae, respectively. Long-termrelativistic jets may be launched by the merger remnant as hinted in X-raylight curves of some SGRBs. The prolonged jets may dissipate their kineticenergy within the radius of the cocoon formed by the jet-ejecta interaction.Then the cocoon supplies seed photons to non-thermal electrons accelerated atthe dissipation region, causing high-energy gamma-ray production through theinverse Compton scattering process. We numerically calculate high-energygamma-ray spectra in such a system using a one-zone and steady-stateapproximation, and show that GeV--TeV gamma-rays are produced with a durationof $10^2-10^5$ seconds. They can be detected by {\it Fermi}/LAT or CTA asgamma-ray counterparts to GWs.

Upscattered Cocoon Emission in Short Gamma-ray Bursts as High-energy Gamma-ray Counterparts to Gravitational Waves