Online algorithm is a well-known computational model. We introduce quantum online algorithms and investigate them with respect to a competitive ratio in two points of view: space complexity and advice complexity. We start with exploring a model with restricted memory and show that quantum online algorithms can be better than classical ones (deterministic or randomized) for sublogarithmic space (memory), and they can be better than deterministic online algorithms without restriction for memory. Additionally, we consider polylogarithmic space case and show that in this case, quantum online algorithms can be better than deterministic ones as well. Another point of view to the online algorithms model is advice complexity. So, we introduce quantum online algorithms with a quantum channel with an adviser. Firstly, we show that quantum algorithms have at least the same computational power as classical ones have. And we give some examples of quantum online algorithms with advice. Secondly, we show that if we allow to use shared entangled qubits (EPR-pairs), then quantum online algorithm can use two times less advise qubits comparing to a classical one. We apply this approach to the well-known Paging Problem.
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