If you are referring to slerp, the matrix version is trivial using the log and exp maps.

    slerp(A, B, t) = exp(t log(B A^T)) A

Something more complex like splines, I am not familiar with the algorithms used for quaternions. I would bet they use Lie algebraic operations, but do they actually use anything specific to quaternions?

Not disagreeing that quaternions have some computational advantages over matrices, but most of the points people repeat as "advantages of quaternions" are really "advantages of proper Lie group/algebra operations over Euler angles" that are independent of matrix vs. quaternion representation.