Nearly all designs of accelerators for heavy ion fusion rely on a velocity (energy) ramp to compress the beam longitudinally from its length in the accelerator to the length required at the target. The size of the velocity ramp is constrained by the longitudinal emittance of the beam. For example, if the longitudinal emittance is 0.05 eV {center_dot} s and we wish to produce a pulse having a width of {+-}2.5 ns at the target, we must supply an energy tilt such that the energy spread at the target is at least {+-}0.05 eV {center_dot} s/2.5 ns = {+-}2 x 10{sup 7} eV. The minimal value of energy spread occurs when the beam has propagated to the point where there is no correlation between the time and energy variables of the beam particles. (In the simple approximation where the boundary of the longitudinal phase space containing the particles is an ellipse, the ellipse is erect at this point, i.e., not tilted with respect to the axes.) In any case, the energy spread can affect focusing. If, for example, the beam kinetic energy is of the order of 5 GeV, a tilt of {+-}2 x 10{sup 7} eV corresponds to a fractional energy spread of 0.004 and it may be possible to focus the beam to the required spot size without using an achromatic optical system. Nevertheless, an optical system that allows larger longitudinal emittance should lead to a less expensive accelerator since the tolerances on acceleration waveforms could be relaxed. Moreover, at lower kinetic energies the problem becomes more serious. If the kinetic energy of our example beam were 1 GeV rather than 5 GeV, the fractional energy spread would be 0.02. This much energy spread would likely produce serious chromatic aberrations leading to an unwanted increase in focal spot size. It is interesting to note that the lower limit on energy spread at the target does not depend on whether the beam is neutralized as it approaches the target. If the beam is not neutralized, it will require a larger initial velocity tilt to overcome longitudinal space-charge forces; but these forces will remove part of the tilt as the beam compresses. Al Maschke suggested that it is possible to reduce the chromatic aberrations by applying a time-dependent transverse focusing correction to the beam upstream of the final focusing lenses [1]. At this point, because of the energy tilt, there is a correlation between longitudinal position in the beam and particle energy. In other words, the average beam energy at the tail of the beam is larger than the average beam energy at the head of the beam. If the beam is completely neutralized as it drifts toward the final focusing lenses, the kinetic energies of the individual particles will remain nearly unchanged during compression. In this case, it is possible, in principle, to apply some 'pre-focusing' to the higher energy particles (those nearer to the tail of the beam) to compensate for their weaker focusing in the final lenses. Although kinetic energies of individual particles are not conserved if the beam is not neutralized, one still expects a positive correlation between the particle energies at the beginning of compression and at the end of compression so correction is still assumed to be possible. It is important that the pulse duration is larger upstream than it is at the final focusing lenses. Larger pulse duration makes it easier, from an engineering standpoint, to supply the power needed to drive the pulsed correction elements. Nevertheless, it still appears impossible or very costly to provide the needed power for some specific cases that have been studied. In the remainder of this paper we ignore this issue and try to determine if there are other fundamental limitations on how well one might correct. We conclude that there are other important limitations.