After beating the main game, watch all the credits all the way through. After they are finished you have the option to save your game. Save it and you will unlock both Hard and Time Attack Mode.

Unlockable	How to Unlock
Hard Mode	After beating the game, you have the option to start a new game in Hard Mode on the title screen.
Time Attack Hard Mode	Beat the game in Hard Mode.
Time Attack Mode	Load your save data after completing the game. Pray infront of any Colossus Statue by pressing the "O" button to enter Time Attack Mode.