A few days ago, the PS4 version of the Evolve Big Alpha was delayed due to the recent Firmware Update 2.0. Today, however, 2K Games finally announced that the alpha is now live for PS4 in an official statement:

After exploring all available solutions, we believe we have addressed the majority of known issues affecting the Evolve Big Alpha on PlayStation 4 following the PS4 2.0 firmware update. The alpha test is now live on PS4 after a gradual roll out to registered participants. Evolve’s closed alpha test is also being extended on Xbox One, PC and PS4 through Tuesday, November 4 at 12:00 PM PST (8:00 PM UTC).

Alpha tests analyze and troubleshoot unfinished products. While internal testing shows the majority of PS4 matches are completed successfully, some players will encounter interruptions and issues affecting the Evolve experience. We ask for your patience and participation as we rigorously test Evolve to ensure a successful launch on February 10, 2015.

We thank everyone helping us test Evolve. Your participation in the Evolve Big Alpha plays a critical role in allowing us to develop the best possible gaming experience.

Additionally, there's a 54 megabytes update to download on PS4. Presumably it's the patch fixing the issued that prevented the alpha from starting on Sony's new console. Due to the issue, 2K has also extended the alpha until November 4th. You can also check out a screenshot of the patch notes posted below to see what was exactly fixed: