  We consider online algorithms for the $k$-server problem on trees. Chrobak
and Larmore proposed a $k$-competitive algorithm for this problem that has the
optimal competitive ratio. However, a naive implementation of their algorithm
has $O(n)$ time complexity for processing each query, where $n$ is the number
of nodes in the tree. We propose a new time-efficient implementation of this
algorithm that has $O(n\log n)$ time complexity for preprocessing and
$O\left(k^2 + k\cdot \log n\right)$ time for processing a query. We also
propose a quantum algorithm for the case where the nodes of the tree are
presented using string paths. In this case, no preprocessing is needed, and the
time complexity for each query is $O(k^2\sqrt{n}\log n)$. When the number of
queries is $o\left(\frac{\sqrt{n}}{k^2\log n}\right)$, we obtain a quantum
speed-up on the total runtime compared to our classical algorithm.
  We also give a simple quantum algorithm to find the first marked element in a
collection of $m$ objects, that works even in the presence of two-sided bounded
errors on the input oracle. It has worst-case complexity $O(\sqrt{m})$. In the
particular case of one-sided errors on the input, it has expected time
complexity $O(\sqrt{x})$ where $x$ is the position of the first marked element.
Compare with previous work, our algorithm can handle errors in the input
oracle.
