  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 NA62 experiment at the CERN SPS will measure
the BR for the charged channel to better than 20%. The BR for the neutral
channel 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. 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 amid an intense
background from soft photons and neutrons in the beam. We present findings from
our design studies, with an emphasis on the challenges faced and the potential
sensitivity for the measurement of BR($K_L\to\pi^0\nu\bar{\nu}$).
