  Quasars represent a brief phase in the life-cycle of most massive galaxies,
but the evolutionary connection between central black holes and their host
galaxies remains unclear. While quasars are active and shining brighter than
the Compton-limit luminosity, their radiation heats the surrounding medium to
the Compton temperature, forming Compton spheres extending to the Str\"omgren
radius of Fe$^{26+}$/He$^{2+}$. After the quasars shut off, their ``afterglow''
can be detected through three signatures: (1) an extended X-ray envelope, with
a characteristic temperature of $\sim 3\times 10^7$ K; (2) Ly$\alpha$ and
Ly$\beta$ lines and the $K-$edge of Fe$^{26+}$; and (3) nebulosity from
hydrogen and helium recombination emission lines. We discuss the possibility of
detecting these signatures using {\em Chandra}, the planned {\em XEUS} mission,
and ground-based optical telescopes. The luminosity and size of quasar
afterglows can be used to constrain the lifetime of quasars.
