MPEJ Volume 5, No.1, 16pp Received: June 19, 1998, Accepted: Jan 6, 1999 I. Guarneri, H. Schulz-Baldes Lower bounds on wave packet propagation by packing dimensions of spectral measures ABSTRACT: We prove that, for any quantum evolution in $\ell^2(\ZZ^D)$, there exist arbitrarily long time scales on which the $q$th moment of the position operator increases at least as fast as a power of time given by $q/D$ times the packing dimension of the spectral measure. Packing dimensions of measures and their connections to scaling exponents and box-counting dimensions are also discussed.