In case anyone else is having trouble finding the beta client by searching the Playstation Store, here's a direct link:
https://store.sonyentertainmentnetwork.com/#!/en-us/games/final-fantasy-xiv-a-realm-reborn-(beta-version)/cid=UP0082-CUSA00288_00-C000030000000000?smcid=ps:show-now:visit-playstation-store:ps-playstation-store
'Purchase' it from there and then head to your library on your console in Playstation Store. You can initiate the download from there.
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It's a 12GB client, just like the last time. I have it about a third of the way downloaded myself. I couldn't find it last night on the PS Store so I assumed it wasn't 'live' yet. Anyone else playing on the PS4 right now?