How can I run a hypothesis test to check that a playlist shuffle is working properly?

Question

Let's say I want to run a hypothesis test for my Apple Music playlist. Let's pretend my playlist has 50 songs for simplicity.

It seems there are two conditions to satisfy:

1. The probability of any song coming up is uniform
2. There is no correlation between songs

Testing for uniformity seems simple enough, we can run an experiment clicking shuffle N times. Then we can do a Chi-squared test to test that the distribution is uniform.

I'm having trouble understanding how I would test for 2) and if that is even the right approach. Is this the same as testing for randomness?

I have done a lot of searching for randomness tests and it seems the Runs test or Diehard tests are common approaches. However, those approaches do not seem intuitive to me in this scenario.