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In the thesis, we use a recently developed tight characterisation of quantumquery complexity, the adversary bound, to develop new quantum algorithms andlower bounds. Our results are as follows:  * We develop a new technique for the construction of quantum algorithms:learning graphs.  * We use learning graphs to improve quantum query complexity of the triangledetection and the $k$-distinctness problems.  * We prove tight lower bounds for the $k$-sum and the triangle sum problems.  * We construct quantum algorithms for some subgraph-finding problems that areoptimal in terms of query, time and space complexities.  * We develop a generalisation of quantum walks that connects electricalproperties of a graph and its quantum hitting time. We use it to construct atime-efficient quantum algorithm for 3-distinctness.

Applications of the Adversary Method in Quantum Query Algorithms