This manuscript illustrates quasi-ideals and minimal quasi-ideals in a ternary semiring. Various characterizations of these ideals have been established. In fact the quasi-simple ternary semirings initiated in terms of quasi-ideals and minimal quasi-ideals. It was shown that if any quasi-ideal is itself a quasi-simple ternary semiring, then it becomes a minimal quasi-ideal. Mathematics Subject classification (2010): 16Y60, 16Y99.