  Quantum phase estimation (QPE) is one of the most important subroutines in
quantum computing. In general applications, current QPE algorithms either
suffer an exponential time overload or require a set of - notoriously quite
fragile - GHZ states. These limitations have prevented so far the demonstration
of QPE beyond proof-of-principles. Here we propose a new QPE algorithm that
scales linearly with time and is implemented with a cascade of Gaussian spin
states (GSS). GSS are renownedly resilient and have been created experimentally
in a variety of platforms, from hundreds of ions up to millions of
cold/ultracold neutral atoms. We show that our protocol achieves a QPE
sensitivity overcoming previous bounds, including those obtained with GHZ
states, and is robustly resistant to several sources of noise and decoherence.
Our work paves the way toward realistic quantum advantage demonstrations of the
QPE, as well as applications of atomic squeezed states for quantum computation.
