We report on the higher-order photon correlations of a high-$beta$ nanolaser under pulsed excitation at room temperature. Using a multiplexed four-element superconducting single photon detector we measured g$^{(n)}(vec{0})$ with $n$=2,3,4. All orders of correlation display partially chaotic statistics, even at four times the threshold excitation power. We show that this departure from coherence and Poisson statistics is due to the quantum fluctuations associated with the small number of dipoles and photons involved in the lasing process.
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