What unlocks Act IV?

Completing any five of the sixteen Legendary Eye quests in Journey To The End opens the last quest of Act III, and entering the End completes it and opens Act IV.

The Eight Sins quest says four Cataclysm bosses are needed, but no boss kill is part of the gate. Getting into the End is the harder half: the stronghold portal needs twelve different Legendary Eyes, one per frame.

See Act IV – Beyond Reality Gate for the details and where each eye comes from.