No. The remastered version does not support cross-save with the PS3 original.
: The game is currently available on the PlayStation Store and through PS Plus Premium subscriptions. Academic Perspectives on "PKG Links" and Piracy god of war iii remastered ps4 pkg link
: One of the most notable upgrades is the visual enhancement. The game now runs at a smoother 60 frames per second, making the combat and exploration even more fluid. The textures have been improved, providing a sharper and more detailed world to explore. god of war iii remastered ps4 pkg link