  Precise measurements of the branching ratios for the flavor-changing neutral
current decays $K\to\pi\nu\bar{\nu}$ can provide unique constraints on CKM
unitarity and, potentially, evidence for new physics. It is important to
measure both decay modes, $K^+\to\pi^+\nu\bar{\nu}$ and
$K_L\to\pi^0\nu\bar{\nu}$, since different new physics models affect the rates
for each channel differently. The goal of the NA62 experiment at the CERN SPS
is to measure the BR for the charged channel to within 10%. For the neutral
channel, the BR has never been measured. We are designing the KLEVER experiment
to measure BR($K_L\to\pi^0\nu\bar{\nu}$) to $\sim$20% using a high-energy
neutral beam at the CERN SPS starting in LHC Run 4. The boost from the
high-energy beam facilitates the rejection of background channels such as
$K_L\to\pi^0\pi^0$ by detection of the additional photons in the final state.
On the other hand, the layout poses particular challenges for the design of the
small-angle vetoes, which must reject photons from $K_L$ decays escaping
through the beam exit amidst an intense background from soft photons and
neutrons in the beam. Background from $\Lambda \to n\pi^0$ decays in the beam
must also be kept under control. We present findings from our design studies
for the beamline and experiment, with an emphasis on the challenges faced and
the potential sensitivity for the measurement of BR($K_L\to\pi^0\nu\bar{\nu}$).
