There exists strong circumstantial evidence from their eccentric orbits that most of the known giant exoplanet systems are the survivors of violent dynamical instabilities. We numerically simulate the evolution of planetary systems around Sun-like stars with three components: (i) an inner disk of planetesimals and planetary embryos, (ii) three giant planets at Jupiter- Saturn distances, and (iii) an outer disk of planetesimals comparable to the primitive Kuiper belt. We calculate the dust production and spectral energy distribution of each system by assuming that each planetesimal particle represents an ensemble of smaller bodies in collisional equilibrium. Our main result is a strong correlation between the presence of terrestrial planets and debris disks. Strong giant planet instabilities that produce very eccentric surviving planets destroy all rocky material in the system, including fully-formed terrestrial planets if the instabilities occur late, and also destroy the icy planetesimal population. Stable or weakly unstable systems allow terrestrial planets to accrete in their inner regions and significant dust to be produced in their outer regions, detectable at midinfrared wavelengths as debris disks. Stars older than 100 Myr with bright cold dust emission (in particular at 70μm) signpost dynamically calm environments that were conducive to efficient terrestrial accretion. Such emission is present around 16% of billion-year old Solar-type stars. We make two predictions. First, eccentric giant planets should be anticorrelated with both debris disks and terrestrial exoplanets. Second, the presence of debris disks and terrestrial exoplanets should be correlated.< Réduire