If I use Gibbs sampling with a Bayesian model, what do I have to check is memoryless?

Right now I am trying to better understand how Bayesian modeling works with just the basics. I found through reading tutorials that some very basic Bayesian models like Bayesian Hierarchical Modeling use something called the "Gibbs sampling algorithm", which is a Markov Chain Monte Carlo Method algorithm.

I know that, if I am going to do anything with Markov Chains, then I have to test a data or parameter violates the assumption of memoryless. However, I am uncertain what exactly do I need to test for memorylessness. Is it a parameter, the dataset, the response variable, a particular residual or error distribution.... what is it that has to be memoryless if you use Gibbs Sampling?

(P.P.S.) I did find a paper on that test's memoryless for Bayesian models but its really confusing. I am a little familiar with how to test memorylessness for basic data, but I am lost on more complex algorithms like Gibbs Sampling.