This work studies the question of quantified derandomization, which was introduced by Goldreich and Wigderson (STOC 2014). The generic quantified derandomization problem is the following: For a circuit class $\mathcal{C}$ and a parameter $B=B(n)$, given a circuit $C\in\mathcal{C}$ with $n$ input bits, decide whether $C$ rejects all of its inputs, or accepts all but $B(n)$ of its inputs. In the current work we consider three settings for this question. In each setting, we bring closer the parameter setting for which we can unconditionally construct relatively fast quantified derandomization algorithms, and the ``threshold'' values (for the parameters) for which any quantified derandomization algorithm implies a similar algorithm for standard derandomization.

For constant-depth circuits, we construct an algorithm for quantified derandomization that works for a parameter $B(n)$ that is only slightly smaller than a ``threshold'' parameter, and is significantly faster than the best currently-known algorithms for standard derandomization. On the way to this result we establish a new derandomization of the switching lemma, which significantly improves on previous results when the width of the formula is small. For constant-depth circuits with parity gates, we lower a ``threshold'' of Goldreich and Wigderson from depth five to depth four, and construct algorithms for quantified derandomization of a remaining type of layered depth-$3$ circuit that they left as an open problem. We also consider the question of constructing hitting-set generators for multivariate polynomials over large fields that vanish rarely, and prove two lower bounds on the seed length of such generators.

Several of our proofs rely on an interesting technique, which we call the randomized tests technique. Intuitively, a standard technique to deterministically find a ``good'' object is to construct a simple deterministic test that decides the set of good objects, and then ``fool'' that test using a pseudorandom generator. We show that a similar approach works also if the simple deterministic test is replaced with a distribution over simple tests, and demonstrate the benefits in using a distribution instead of a single test.

A final version.
* A gap in the proof of Proposition 39 is filled.
* There are minor changes in notation and exposition.

This work studies the question of quantified derandomization, which was introduced by Goldreich and Wigderson (STOC 2014). The generic quantified derandomization problem is the following: For a circuit class $\mathcal{C}$ and a parameter $B=B(n)$, given a circuit $C\in\mathcal{C}$ with $n$ input bits, decide whether $C$ rejects all of its inputs, or accepts all but $B(n)$ of its inputs. In the current work we consider three settings for this question. In each setting, we bring closer the parameter setting for which we can unconditionally construct relatively fast quantified derandomization algorithms, and the "threshold" values (for the parameters) for which any quantified derandomization algorithm implies a similar algorithm for standard derandomization.

For {\bf constant-depth circuits}, we construct an algorithm for quantified derandomization that works for a parameter $B(n)$ that is only slightly smaller than a "threshold" parameter, and is significantly faster than the best currently-known algorithms for standard derandomization. On the way to this result we establish a new derandomization of the switching lemma, which significantly improves on previous results when the width of the formula is small. For {\bf constant-depth circuits with parity gates}, we lower a "threshold" of Goldreich and Wigderson from depth five to depth four, and construct algorithms for quantified derandomization of a remaining type of layered depth-3 circuit that they left as an open problem. We also consider the question of constructing hitting-set generators for multivariate {\bf polynomials over large fields that vanish rarely}, and prove two lower bounds on the seed length of such generators.

Several of our proofs rely on an interesting technique, which we call the randomized tests technique. Intuitively, a standard technique to deterministically find a "good" object is to construct a simple deterministic test that decides the set of good objects, and then "fool" that test using a pseudorandom generator. We show that a similar approach works also if the simple deterministic test is replaced with a distribution over simple tests, and demonstrate the benefits in using a distribution instead of a single test.

An improvement in the parameters of the main theorem for constant-depth circuits, which is obtained by a slightly nicer construction/proof; various minor corrections.

This work studies the question of quantified derandomization, which was introduced by Goldreich and Wigderson (STOC 2014). The generic quantified derandomization problem is the following: For a circuit class $\mathcal{C}$ and a parameter $B=B(n)$, given a circuit $C\in\mathcal{C}$ with $n$ input bits, decide whether $C$ rejects all of its inputs, or accepts all but $B(n)$ of its inputs. In the current work we consider three settings for this question. In each setting, we bring closer the parameter setting for which we can unconditionally construct relatively fast quantified derandomization algorithms, and the "threshold" values (for the parameters) for which any quantified derandomization algorithm implies a similar algorithm for standard derandomization.

For constant-depth circuits, we construct an algorithm for quantified derandomization that works for a parameter $B(n)$ that is only slightly smaller than a "threshold" parameter, and is significantly faster than the best currently-known algorithms for standard derandomization. On the way to this result we establish a new derandomization of the switching lemma, which significantly improves on previous results when the width of the formula is small. For constant-depth circuits with parity gates, we lower a
"threshold" of Goldreich and Wigderson from depth five to depth four, and construct algorithms for quantified derandomization of a remaining type of layered depth-$3$ circuit that they left as an open problem. We also consider the question of constructing hitting-set generators for multivariate polynomials over large fields that vanish rarely, and prove two lower bounds on the seed length of such generators.

Several of our proofs rely on an interesting technique, which we call the randomized tests technique. Intuitively, a standard technique to deterministically find a "good" object is to construct a simple deterministic test that decides the set of good objects, and then "fool" that test using a pseudorandom generator. We show that a similar approach works also if the simple deterministic test is replaced with a distribution over simple tests, and demonstrate the benefits in using a distribution instead of a single test.

A significant revision, which includes a new and more general main theorem for constant-depth circuits, a new derandomization of the switching lemma, and a clearer exposition of the randomized tests technique.

Goldreich and Wigderson (STOC 2014) initiated a study of quantified derandomization, which is a relaxed derandomization problem: For a circuit class $\mathcal{C}$ and a parameter $B=B(n)$, the problem is to decide whether a circuit $C\in\mathcal{C}$ rejects all of its inputs, or accepts all but $B(n)$ of its inputs.

In this work we make progress on several frontiers that they left open. Specifically, for constant-depth circuits, we construct an algorithm for quantified derandomization that is significantly faster than the best currently-known algorithms for standard derandomization, and works for a parameter $B(n)$ that is only slightly smaller than a ``barrier'' parameter that was shown by Goldreich and Wigderson. For constant-depth circuits with parity gates, we tighten a ``barrier'' of Goldreich and Wigderson (from depth five to depth four), and construct algorithms for quantified derandomization of a remaining type of layered depth-$3$ circuit that they did not handle and left as an open problem (i.e., circuits with a top $\oplus$ gate, a middle layer of $\land$ gates, and a bottom layer of $\oplus$ gates).

In addition, we extend Goldreich and Wigderson's study of multivariate polynomials that vanish rarely to the setting of large finite fields. We prove two lower bounds on the seed length of hitting-set generators for polynomials over large fields that vanish rarely. As part of the proofs, we show a form of ``error reduction'' for polynomials (i.e., a reduction of the task of hitting arbitrary polynomials to the task of hitting polynomials that vanish rarely) that causes only a mild increase in the degree.