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In quantum phase estimation, the Heisenberg limit provides the ultimateaccuracy over quasi-classical estimation procedures. However, realizing thislimit hinges upon both the detection strategy employed for output measurementsand the characteristics of the input states. This study delves into quantumphase estimation using $s$-spin coherent states superposition. Initially, wedelve into the explicit formulation of spin coherent states for a spin $s=3/2$.Both the quantum Fisher information and the quantum Cramer-Rao bound aremeticulously examined. We analytically show that the ultimate measurementprecision of spin cat states approaches the Heisenberg limit, where uncertaintydecreases inversely with the total particle number. Moreover, we investigatethe phase sensitivity introduced through operators $e^{i\zeta{S}_{z}}$,$e^{i\zeta{S}_{x}}$ and $e^{i\zeta{S}_{y}}$, subsequently comparing theresultants findings. In closing, we provide a general analytical expression forthe quantum Cramer-Rao boundary applied to these three parameter-generatingoperators, utilizing general $s$-spin coherent states. We remarked thatattaining Heisenberg-limit precision requires the careful adjustment ofinsightful information about the geometry of $s$-spin cat states on the Blochsphere. Additionally, as the number of $s$-spin increases, the Heisenberg limitdecreases, and this reduction is inversely proportional to the $s$-spin number.

Achieving quantum metrological performance and exact Heisenberg limit precision through superposition of $s$-spin coherent states