  Cram\'er-Rao constitutes a crucial lower bound for the mean squared error of
an estimator in frequentist parameter estimation, albeit paradoxically
demanding highly accurate prior knowledge of the parameter to be estimated.
Indeed, this information is needed to construct the optimal unbiased estimator,
which is highly dependent on the parameter. Conversely, Bhattacharyya bounds
result in a more resilient estimation about prior accuracy by imposing
additional constraints on the estimator. Initially, we conduct a quantitative
comparison of the performance between Cram\'er-Rao and Bhattacharyya bounds
when faced with less-than-ideal prior knowledge of the parameter. Furthermore,
we demonstrate that the $n^{th}$order classical and quantum Bhattacharyya
bounds cannot be computed -- given the absence of estimators satisfying the
constraints -- under specific conditions tied to the dimension $m$ of the
discrete system. Intriguingly, for a system with the same dimension $m$, the
maximum non-trivial order $n$ is $m-1$ in the classical case, while in the
quantum realm, it extends to $m(m+1)/2-1$. Consequently, for a given system
dimension, one can construct estimators in quantum systems that exhibit
increased robustness to prior ignorance.
