The Karolyhazy uncertainty relation states that if a device is used to measure a length l, there will be a minimum uncertainty \(\delta l\) in the measurement, given by \((\delta l)^3 \sim L_{\mathrm {P}}^2 l\). This is a consequence of combining the principles of quantum mechanics and general relativity. In this letter we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space–time–matter. We use this relation to define a space–time–matter (STM) foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time-scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever classical space–time background is not available to describe a quantum system. Space–time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.

Karolyhazy不确定性关系指出，如果使用设备测量长度l，则测量中将存在最小不确定性\（\ delta l \），由\（（\ delta l）^ 3 \ sim L _ {\ mathrm {P}} ^ 2 l \）。这是量子力学原理和广义相对论相结合的结果。在这封信中，我们说明了这种关系如何在我们的量子引力方法中以自下而上的方式从时空物质原子的矩阵动力学中产生。我们用这种关系来定义普朗克尺度的时空泡沫（STM），并认为我们的理论是全息的。通过在比普朗克时间更大的时间尺度上进行粗粒化，可以获得量子引力定律。量子引力不是普朗克尺度的现象；相反，只要没有经典的时空背景来描述量子系统，它就会发挥作用。时空和经典广义相对论源于高度纠缠的量子引力系统中的自发局域性。Karolyhazy关系继续存在于涌现理论中。