  We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in
$O(n^{1-1/k})$ rounds in the \CONGEST{} model by a randomized Monte-Carlo
distributed algorithm with one-sided error probability $1/3$. This matches the
best round-complexities of previously known algorithms for $k\in\{2,3,4,5\}$ by
Drucker et al. [PODC'14] and Censor-Hillel et al. [DISC'20], but improves the
complexities of the known algorithms for $k>5$ by Eden et al. [DISC'19], which
were essentially of the form $\tilde O(n^{1-2/k^2})$. Our algorithm uses
colored BFS-explorations with threshold, but with an original \emph{global}
approach that enables to overcome a recent impossibility result by Fraigniaud
et al. [SIROCCO'23] about using colored BFS-exploration with \emph{local}
threshold for detecting cycles.
  We also show how to quantize our algorithm for achieving a round-complexity
$\tilde O(n^{\frac{1}{2}-\frac{1}{2k}})$ in the quantum setting for deciding
$C_{2k}$ freeness. Furthermore, this allows us to improve the known quantum
complexities of the simpler problem of detecting cycles of length \emph{at
most}~$2k$ by van Apeldoorn and de Vos [PODC'22]. Our quantization is in two
steps. First, the congestion of our randomized algorithm is reduced, to the
cost of reducing its success probability too. Second, the success probability
is boosted using a new quantum framework derived from sequential algorithms,
namely Monte-Carlo quantum amplification.
