What is the Space Complexity of Tail Recursive Quicksort?
If you either use a compiler that supports tail recursion, or make the small change not to use recursion for the larger partition at all, the recursion depth depends on the sizes of the smaller intervals. 

The worst case for recursion depth is the case that each partition divides your subarray exactly in half, making the depth $log_2 (n)$. The best case for recursion depth is the case that each partition divides your subarray into n-1 and 1 items, recursion depth = 1 (obviously the absolutely worst case for execution time). 

So this is a bit paradoxical: Best case space complexity = worst case time complexity. But on the other hand, worst case space complexity = log n is so little that nobody cares about the space complexity. As long obviously as people don't make the mistake of sorting the larger partition first. 

Without eliminating the recursion for the larger half, the worst case for recursion depth is n, and the order of the calls doesn’t matter. 

Usually you can’t make an assumption that values would be different, so the quality of your partition function is important. Some will misbehave with many identical values.