We give a lower bound of ?(?n) on the unambiguous randomised parity-query complexity of the approximate majority problem – that is, on the lowest randomised parity-query complexity of any function over {0,1}? whose value is "0" if the Hamming weight of the input is at most n/3, is "1" if ... more >>>

We define and study the model of patterned non-determinism in bipartite communication complexity, denoted by $PNP^{X\leftrightarrow Y}$.
It generalises the known models $UP^{X\leftrightarrow Y}$ and $FewP^{X\leftrightarrow Y}$ through relaxing the constraints on the witnessing structure of the underlying $NP^{X\leftrightarrow Y}$-protocol.
It is shown that for the case of total functions ... more >>>

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is ... more >>>

Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $\R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. ... more >>>

Two parties wish to carry out certain distributed computational tasks, and they are given access to a source of correlated random bits.
It allows the parties to act in a correlated manner, which can be quite useful.
But what happens if the shared randomness is not perfect?

In this work, ... more >>>

We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any ... more >>>

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}_1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the ... more >>>

We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that ... more >>>

We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case.

First, we demonstrate a hierarchy of modes of shared ... more >>>

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,\ldots,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,\ldots,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,\ldots,Y_r$.

We introduce a new type of cryptographic primitive that we call hiding fingerprinting. No classical fingerprinting scheme is hiding. We construct quantum hiding fingerprinting schemes and argue their optimality.

We prove that NP$\ne$coNP and coNP$\nsubseteq$MA in the number-on-forehead model of multiparty communication complexity for up to $k=(1-\epsilon)\log n$ players, where $\epsilon>0$ is any constant. Specifically, we construct a function $F:(\zoon)^k\to\zoo$ with co-nondeterministic
complexity $O(\log n)$ and Merlin-Arthur
complexity $n^{\Omega(1)}$.
The problem was open for $k\geq3$.

We give the first exponential separation between quantum and
classical multi-party
communication complexity in the (non-interactive) one-way and
simultaneous message
passing settings.
For every k, we demonstrate a relational communication problem
between k parties
that can be solved exactly by a quantum simultaneous message passing
protocol of
cost ... more >>>

We demonstrate a two-player communication problem that can be solved in the one-way quantum model by a 0-error protocol of cost O(log n) but requires exponentially more communication in the classical interactive (two-way) model.

We give an exponential separation between one-way quantum and classical communication complexity for a Boolean function. Earlier such a separation was known only for a relation. A very similar result was obtained earlier but independently by Kerenidis and Raz [KR06]. Our version of the result gives an example in the ... more >>>