We study the notion of "instance compressibility" of NP problems [Harnik-Naor06], closely related to the notion of kernelization in parameterized complexity theory [Downey-Fellows99, Flum-Grohe06, Niedermeier06]. A language $L$ in NP is instance compressible if there
is a polynomial-time computable function $f$ and a set $A$ such that
for each instance ... more >>>

Among others, refining the methods of [Fortnow and Santhanam, ECCC Report TR07-096] we improve a result of this paper and show for any parameterized problem with a ``linear weak OR'' and with NP-hard underlying classical problem that there is no polynomial reduction from the problem to itself that assigns to ... more >>>

We show that, in the black-box setting, the behavior of quantum polynomial-time (${BQP}$) can be remarkably decoupled from that of classical complexity classes like ${NP}$. Specifically:

-There exists an oracle relative to which ${NP}^{{BQP}}\not \subset {BQP}^{{PH}}$, resolving a 2005 problem of Fortnow. Interpreted another way, we show that ${AC^0}$ circuits ... more >>>

We introduce the entangled quantum polynomial hierarchy $QEPH$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $QEPH$ collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. ... more >>>

TFNP studies the complexity of total, verifiable search problems, and represents the first layer of the total function polynomial hierarchy (TFPH). Recently, problems in higher levels of the TFPH have gained significant attention, partly due to their close connection to circuit lower bounds. However, very little is known about the ... more >>>