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In this paper, we consider the design of multi-stream demodulators for multiple-input multiple-output (MIMO) systems. Our proposed MIMO demodulator is based on the stack tree-search strategy and provides soft-outputs in the form of exact max-log log-likelihood ratios (max-log LLRs). We prove that our proposed demodulator is optimally efficient in that it visits the least number of nodes among all optimal demodulators in a class of tree-search-based demodulators. We conduct a comprehensive complexity analysis of such optimally efficient demodulators. We also identify key parameters associated with our proposed demodulator that can be controlled to realize near max-log performance with substantially reduced complexity, as demonstrated via simulations.