You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
We completely characterise the complexity in the decision tree model of computing composite relations of the form h = g(f^1,...,f^n), where each relation f^i is boolean-valued. Immediate corollaries include a direct sum theorem for decision tree complexity and a tight characterisation of the decision tree complexity of iterated boolean functions.

Cited Abstract(s):
[1]: We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f(g,...,g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It further shows that span programs are equivalent to quantum query algorithms.

[2]: The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations interms of weight schemes, eigen values, and Kolmogorov complexity. All these formulations rely on the principlethat if an algorithm successfully computes a function then, in particular, itis able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function. This new method, which we call ADV+-, has all the advantages ofthe old: it is a lower bound on bounded-error quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect tofunction composition. Moreover ADV+- is always at least as large as the adversary method ADV, and we show an example of a monotone function forwhich ADV+-(f)=Omega(ADV(f)1.098). We also give examples showing that ADV+- does not face limitations of ADV like the certificate complexity barrier and the property testing barrier.

[3]: The quantum adversary method is a very versatile method for proving lower bounds on quantum algorithms. It has many equivalent formulations, yields tight bounds for many computational problems, and has natural connections to classical lower bounds. One of its formulations is in terms of the spectral norm of some matrices. We define a weighted version of this spectral method and show that it possesses useful composition properties. The results generalize and unify previously known composition properties of adversary methods.

[4]: The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coincide with the general adversary lower bound. Second, we give a quantum algorithm for evaluating span programs with only a logarithmic query overhead on the witness size. The first result is motivated by a quantum algorithm for evaluating composed span programs. The algorithm is known to be optimal for evaluating a large class of formulas. The allowed gates include all constant-size functions for which there is an optimal span program. So far, good span programs have been found in an ad hoc manner, and the SDP automates this procedure. Surprisingly, the SDP's value equals the general adversary bound. A corollary is an optimal quantum algorithm for evaluating "balanced" formulas over any finite boolean gate set. The second result extends span programs' applicability beyond the formula-evaluation problem. We extend the analysis of the quantum algorithm for evaluating span programs. The previous analysis shows that a corresponding bipartite graph has a large spectral gap, but only works when applied to the composition of constant-size span programs. We show generally that properties of eigenvalue-zero eigenvectors in fact imply an "effective" spectral gap around zero. A strong universality result for span programs follows. A good quantum query algorithm for a problem implies a good span program, and vice versa. Although nearly tight, this equivalence is nontrivial. Span programs are a promising model for developing more quantum algorithms.
Similar, but more complicated, composition theorems to the present work have been proven for quantities which are important in the study of quantum query complexity. H yer, Lee and S palek [3] proved that the so-called non-negative weight adversary bound (which we will not define here; see [3] for details) obeys a composition theorem. Interestingly, their proof is based on generalising the query model to weighted queries, similarly to the approach taken here. A composition theorem was later proven for the general adversary bound (in two parts; a lower bound by H yer, Lee and S palek [2] , and an upper bound by Reichardt [4] ), which was then used by Reichardt to infer a composition theorem for quantum query complexity [1] .