Okay guys. We have to do these fights were they each have their full power. Crota would have all his powers like in his dimension and Atheon would have all his powers as if he were in the Vault of Glass. Now with those considered, Atheon would obviously win because he can just harass Crota and his minions by sending them back in time. Even though we know they would be able to kill the Oracles cause Crota would just spawn some minions to shoot them. However I don't think any of them would know how to use the relic so it would be hopeless for them.
Another way I like to view this is who would survive the longest in each others raids, Crota or Atheon. I feel like Atheon would be able to make it to Crota but just unable to damage him. Assuming Crota got the door open he would only get to the gorgons then be willed from existence. So Atheon just wins in all cases. How could you fight something with control over time?
Don't bring up Atheon having no powers outside of vault please. We are only talking about if each had his full power at all times.
Edit: Just kidding on them killing oracles because if they can't use relic they can't see so they would fall off edge or succumb to oracles.
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Crota would get ported to Mars outside the containment bubble and get instakilled. Then he'd bitch on the forums, making a clever reference to VoG standing for Vault of Glitch. Then he'd say "nerf vex" and start watching Sister Wives.