  We study the complexity of holomorphic isometries and proper maps from the
complex unit ball to type IV classical domains. We investigate on degree
estimates of holomorphic isometries and holomorphic maps with minimum target
dimension. We also construct a real-parameter family of mutually inequivalent
holomorphic isometries from the unit ball to type IV domains. We also provide
examples of non-isometric proper holomorphic maps from the complex unit ball to
classical domains.
