  In this paper, we develop reliable a posteriori error estimates for numerical
approximations of scalar hyperbolic conservation laws in one space dimension.
  Our methods have no inherent small-data limitations and are a step towards
error control of numerical schemes for systems. We are careful not to appeal to
the Kruzhkov theory for scalar conservation laws. Instead, we derive novel
quantitative stability estimates that extend the theory of shifts, and in
particular, the framework for proving stability first developed by the second
author and Vasseur. This is the first time this methodology has been used for
quantitative estimates.
  We work entirely within the context of the theory of shifts and
$a$-contraction, techniques which adapt well to systems. In fact, the stability
framework by the second author and Vasseur has itself recently been pushed to
systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for $2\times 2$
conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022].
  Our theoretical findings are complemented by a numerical implementation in
MATLAB and numerical experiments.
