  We study possible advantages of randomized and quantum computing over
deterministic computing for scalar initial-value problems for ordinary
differential equations of order k. For systems of equations of the first order
this question has been settled modulo some details in \cite{Kacewicz05}. A
speed-up over deterministic computing shown in \cite{Kacewicz05} is related to
the increased regularity of the solution with respect to that of the right-hand
side function. For a scalar equation of order k (which can be transformed into
a special system of the first order), the regularity of the solution is
increased by k orders of magnitude. This leads to improved complexity bounds
depending on k for linear information in the deterministic setting, see
\cite{Szczesny05}. This may suggest that in the randomized and quantum settings
a speed-up can also be achieved depending on k.
  We show in this paper that a speed-up dependent on k is not possible in the
randomized and quantum settings. We establish lower complexity bounds, showing
that the randomized and quantum complexities remain at the some level as for
systems of the first order, no matter how large k is. Thus, the algorithms from
\cite{Kacewicz05} remain (almost) optimal, even if we restrict ourselves to a
subclass of systems arising from scalar equations of order k.
