Uncharted 4: A Thief’s End close beta is live for people who own the Uncharted Collection. It’s a separate thing and does not require the disc once you start the download. You can even rent the Collection.
However, if you don’t want to do any of that. Find a friend who is playing the beta currently and start share play. This will ensure that the beta appears in your recently played list. This will allow you to download the beta.
Next you need to tell your friend to log into your account and create a save data in the Uncharted Collection and upload it to cloud.
Download the save file from cloud and voila, you can now play Uncharted 4 beta on your PS4.
The beta has some matchmaking issues currently so we recommend waiting before Naughty Dog patches it. They are aware of the issue.
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