I can't find any source which says whether quantum or laser gyroscopes are more accurate. I'm also wondering what the advantages and disadvantages are of each technology.
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Many use the terms accuracy, precision, and sensitivity interchangeably when they are different but most often taken to mean sensitivity. I will assume by accurate you mean sensitivity. I will also assume you are referring to atom gyroscopes as quantum as light-based gyroscopes can, in principle also be quantum.
This is not a straightforward question to answer, simply because the devices are so different. For example, we shall look at an all things equal comparison of light and atom-based gyros. The operational principle behind these gyros is the Sagnac effect, for light and atoms propagating around a single loop the phase accumulated due to a rotation is
$$\Phi_L=\frac{8\pi\Omega A}{\lambda c}, \qquad\text{and}\qquad\Phi_A=\frac{4m\Omega A}{\hbar},$$ where $\Omega$ is the rotation being measured $A$ the area enclosed $\lambda$ the wavelength of light and $m$ the mass of the atom. For optical gyroscopes operating with visible light and an atom gyro using $^{87}$Rb, the ratio of the two accumulated phases is $\Phi_A/\Phi_L\approx10^{10}$. This is a huge increase in sensitivity, and this direct comparison is the current drive behind the interest of making atom-based interferometers in general. However, some things have been assumed in this comparison that aren't appropriate for actual applications. The area enclosed was equated, for optical gyroscopes the area enclosed can order of magnitude larger than atom-based gyros. The reason for this is maintaining coherence and achieving the propagation. Another is particle flux, light-based gyros will have a far higher flux of photons than atom-based gyros.
All that said, there is a drive to get atom-based gyros up and running. This is because, in the long run with advancing atom trapping and coherence control techniques, atom-gyros are set to outperform light-based gyros. Although in practice it might be beneficial to have a hybrid gyroscope as atom-gyros have a dead-time (atoms need to be trapped and cooled first before the rotation measurement), the superior short-term sensitivity of an atom-based gyro can be used together with the continuous-mode operation of light-based gyro.
