We consider the problem of design and analysis of optimal L_\infty (minmax) multi-resolution scalar quantizers (MRSQ). The overall multi-resolution L_\infty distortion of an MRSQ is defined to be a weighted sum of L_\infty distortions over all refinement levels of the MRSQ. The weight for a refinement level usually denotes the probability that the MRSQ will operate at that level (rate). An interesting relation of the problem to the design of optimal binary prefix codes under a code cell contiguity constraint is established. Lower bounds for the overall multi-resolution L_\infty distortion are derived based on this relation. Provably optimal as well as fast, near optimal algorithms are also developed for practically interesting scenarios. Furthermore, the performance penalty incurred by making a scalable quantizer embedded (progressively refinable) is analyzed. It is shown that constraining the quantizers to be embedded would on average increase the L_\infty quantization error by at least 44%.