  In the $k$-connected directed Steiner tree problem ($k$-DST), we are given an
$n$-vertex directed graph $G=(V,E)$ with edge costs, a connectivity requirement
$k$, a root $r\in V$ and a set of terminals $T\subseteq V$. The goal is to find
a minimum-cost subgraph $H\subseteq G$ that has $k$ internally disjoint paths
from the root vertex $r$ to every terminal $t\in T$.
  In this paper, we show the approximation hardness of $k$-DST for various
parameters, which thus close some long-standing open problems.
  - $\Omega\left(|T|/\log |T|\right)$-approximation hardness, which holds under
the standard assumption $\mathrm{NP}\neq \mathrm{ZPP}$. The inapproximability
ratio is tightened to $\Omega\left(|T|\right)$ under the Strongish Planted
Clique Hypothesis [Manurangsi, Rubinstein and Schramm, ITCS 2021].
  The latter hardness result matches the approximation ratio of $|T|$ obtained
by a trivial approximation algorithm, thus closing the long-standing open
problem.
  - $\Omega\left(\sqrt{2}^k / k\right)$-approximation hardness for the general
case of $k$-DST under the assumption $\mathrm{NP}\neq\mathrm{ZPP}$. This is the
first hardness result known for survivable network design problems with an
inapproximability ratio exponential in $k$.
  - $\Omega\left((k/L)^{L/4}\right)$-approximation hardness for $k$-DST on
$L$-layered graphs for $L\le O\left(\log n\right)$. This almost matches the
approximation ratio of $O(k^{L-1}\cdot L \cdot \log |T|)$ achieving in
$O\left(n^L\right)$-time due to Laekhanukit [ICALP`16].
