  For sequences $\alpha \equiv \{\alpha_n\}_{n=0}^{\infty}$ of positive real
numbers, called weights, we study the weighted shift operators $W_{\alpha}$
having the property of moment infinite divisibility ($\mathcal{MID}$); that is,
for any $p > 0$, the Schur power $W_{\alpha}^p$ is subnormal. We first prove
that $W_{\alpha}$ is $\mathcal{MID}$ if and only if certain infinite matrices
$\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are conditionally positive
definite (CPD). Here $\gamma$ is the sequence of moments associated with
$\alpha$, $M_{\gamma}(0),M_{\gamma}(1)$ are the canonical Hankel matrices whose
positive semi-definiteness determines the subnormality of $W_{\alpha}$, and
$\log$ is calculated entry-wise (i.e., in the sense of Schur or Hadamard).
Next, we use conditional positive definiteness to establish a new bridge
between $k$--hyponormality and $n$--contractivity, which sheds significant new
light on how the two well known staircases from hyponormality to subnormality
interact. As a consequence, we prove that a contractive weighted shift
$W_{\alpha}$ is $\mathcal{MID}$ if and only if for all $p>0$, $M_{\gamma}^p(0)$
and $M_{\gamma}^p(1)$ are CPD.
