The nature of intermittency in turbulent plasma transport-whether bursts occur as independent events or exhibit memory-is a fundamental question with direct implications for confinement prediction and control. Conventional empirical time-series diagnostics, such as autocorrelation functions and power spectra, are second-order measures and, when used alone, may not fully characterize statistical dependence in intermittent, non-Gaussian plasma fluctuations dominated by nonlinear bursty avalanche-like events. Here we employ information-theoretic tools-block entropy, excess entropy, complexity index, and mutual information-to systematically probe temporal correlations in fluctuation-induced particle flux time series from the Santander Linear Plasma Machine. Experimental data are first converted into binary event/calm sequences via thresholding. Analysis of block entropy reveals that the complexity index grows with block length and remains unsaturated for the outermost radial positions, indicating significant temporal memory and structure. At the plasma edge, mutual information decays slowly and remains detectable over many consecutive bursty events, indicating that information propagates across the avalanche sequence far into the future. This provides a direct evidence that edge plasma turbulence exhibits non-renewal, long-range memory in burst occurrence: the timing of each large event contains predictive information about avalanches many events ahead. The complexity index and quiet-time mutual information together offer sensitive, model-free, quantitative markers of proximity to marginal stability and confinement regime transitions.