Does Mersenne Twister generate homogeneous numbers? I have been assigned to generate 15000+ homogeneous and uniform random numbers for a work.

I have been searching if Mersenne Twister generate homogeneous distribution of numbers but I don't find nothing. I have found a lot of test and proprieties but I do not know if any of those proprieties/tests refers to the homogeneity.

Can you explain me if the algorithm generates homogeneous numbers, why and what are the proprieties involved in?

By homogeneity I think that I mean this.
I think the Mersenne twister satisfies the requirement for homogeneity, since it produces numbers that are equally distributed (with 32-bit accuracy) in up to 623 dimensions.

But in a cryptography context, it could hardly be described as "uniform", because you only need to observe 624 outputs in order to predict all the following values with 100% accuracy.

I would recommend using a cryptographically secure PRNG such as AES-CTR instead. This should provide you with at least 100 MB of random data per second.