  Many-particle entanglement is a key resource for achieving the fundamental
precision limits of a quantum sensor. Optical atomic clocks, the current
state-of-the-art in frequency precision, are a rapidly emerging area of focus
for entanglement-enhanced metrology. Augmenting tweezer-based clocks featuring
microscopic control and detection with the high-fidelity entangling gates
developed for atom-array information processing offers a promising route
towards leveraging highly entangled quantum states for improved optical clocks.
Here we develop and employ a family of multi-qubit Rydberg gates to generate
Schr\"odinger cat states of the Greenberger-Horne-Zeilinger (GHZ) type with up
to 9 optical clock qubits in a programmable atom array. In an atom-laser
comparison at sufficiently short dark times, we demonstrate a fractional
frequency instability below the standard quantum limit using GHZ states of up
to 4 qubits. However, due to their reduced dynamic range, GHZ states of a
single size fail to improve the achievable clock precision at the optimal dark
time compared to unentangled atoms. Towards overcoming this hurdle, we
simultaneously prepare a cascade of varying-size GHZ states to perform
unambiguous phase estimation over an extended interval. These results
demonstrate key building blocks for approaching Heisenberg-limited scaling of
optical atomic clock precision.
