This volume contains a collection of studies in the areas of complexity theory and property testing. The 21 pieces of scientific work included were conducted at different times, mostly during the last decade. Although most of these works have been cited in the literature, none of them was formally published before. Within complexity theory the topics include constant-depth Boolean circuits, explicit construction of expander graphs, interactive proof systems, monotone formulae for majority, probabilistically checkable proofs (PCPs), pseudorandomness, worst-case to average-case reductions, and zero-knowledge proofs. Within property testing the topics include distribution testing, linearity testing, lower bounds on the query complexity (of property testing), testing graph properties, and tolerant testing. A common theme in this collection is the interplay between randomness and computation.

The focus of this monograph is the question of quantified derandomization. Does derandomization of probabilistic algorithms become easier if we only want to derandomize algorithms that err with extremely small probability? How small does this probability need to be in order for the problem's complexity to be affected? In this comprehensive survey of the topic, the author describes the body of knowledge accumulated since the question was first raised in 2014, focusing on the following directions: 1) Hardness vs "quantified" randomness; 2) Improving on the brute-force algorithm for quantified derandomization implies breakthrough circuit lower bounds; 3) Unconditional algorithms for quantified derandomization of low-depth circuits and formulas; 4) Arithmetic quantified derandomization; 5) Limitations of certain black-box techniques and pseudoentropy. Written for researchers, this monograph provides the readers with a concise overview of all known results, but the author also shows several results that are either new or are strengthenings of others. The author also offers a host of concrete challenges and open questions surrounding quantified derandomization.