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We show that for each $k\in \mathbb{N}$, a link $L\subset S^3$ bounds adegree $k$ Whitney tower in the 4-ball if and only if it is\emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from theunlink by a finite sequence of concordances and degree $k$ clasper surgeries.In our construction the trees associated to the Whitney towers coincide withthe trees associated to the claspers.  As a corollary to our previous obstruction theory for Whitney towers in the4-ball, it follows that the $C_k$-concordance filtration of links is classifiedin terms of Milnor invariants, higher-order Sato-Levine and Arf invariants.  Using a new notion of $k$-repeating twisted Whitney towers, we also classifya natural generalization of the notion of link homotopy, called twisted\emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and$k$-repeating Arf invariants.

Clasper Concordance, Whitney towers and repeating Milnor invariants