You can always check Windows Logs in Event Viewer
for any events associated with that program.
But it is almost impossible to find out what you want post factum.
It also depends on a program, depends on many factors.
My be this program make its own logs. May be this program keeps settings in a file and saves it every time it is terminated. In this case we can check date of a file, pretty much the same with torrent client or downloads manager. Prefetch file might give some info as well.
Please give us more specific details to analyze.