Which empire was ruled from Constantinople for a thousand years?

The Byzantine Empire, which is a name it never used. Its people called themselves Romaioi - Romans - and considered their state the continuing Roman Empire, which by any reasonable test it was: unbroken institutional descent from Augustus, in the eastern half, for another thousand years after the west collapsed. The word Byzantine was coined by a German historian in 1557, a century after the empire ended.

That naming has consequences beyond pedantry. Calling it Byzantine let western European historians treat the Roman Empire as ending in 476 and the medieval east as a separate, declining, faintly ridiculous thing - which is where the adjective byzantine, meaning needlessly complex and devious, comes from. A state that lasted a millennium, kept Roman law, preserved Greek literature and repeatedly stopped invasions reaching Europe is remembered mainly as a byword for bureaucracy.

It also had Greek fire, one of history's genuinely lost technologies - a liquid incendiary sprayed from ships that burned on water and could not be put out. The formula was a state secret restricted to a few families, and it was lost so completely that nobody has reconstructed it with confidence.